1131calcprobonly moodle

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Revision questions

Inequalities and Absolute Values

1. Sketch the set of points (x, y) which satisfy the following relations.

a) 0y2x and 0x2 b) y/2x2 and 0y4

2. Solve

a) x(x 1)> 0 b) (x 1)(x 2)< 0 c) 1x >12 d) 11x > 123. Solve

a)x+ 13 c) 3x+ 2


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Limits of some Rational Functions

8. Find

a) limx2x2

5x+ 6

x 2 b) limx2x2

5x+ 6

2x2 3x 2c) lim

12 0.8 0.2

1 d) limx11 x41 x

Simple Differentiation Tangents and Normals

9. Find the derivative of each of the following functions.

a) f(x) = (2x+ 5)3 b) g(x) = sin3 x c) h(x) = cos(x3)

d) f(t) =

t2 4 e) g(t) = tt24 f) h(x) =

x+ex+

g) f() = tan h) g(x) =x cos2x i) f(x) = ex2/2

j) f(x) = x ln x

10. Find the equation of the tangent and the equation of the normal to each of the followingcurves.

a) y= 4x+ 1x at the point (1, 5)

b) y= cos x

1 sin x at the point where x= 6

Stationary Points

11. Locate and identify the stationary points for

a) y= 2x3 9x2 + 12x 3 b) y= x1+x2

c) y= e2x(1 x) d) y= xex

e) y= xnex for n Z, n2 f) y= ln xxg) y= 4x3 x4 h) y= x+ cos x

12. The slope of the curve y = f(x) is given by

dy

dx=x2(2x 1)(x 1)

Determine the nature of the stationary points.

13. The slope of the curve y = f(x) is

dy

dx = 3(x 1)2(x 2)3(x 3)4(x 4)

For what value or values ofx does y have

a) a local maximum? b) a local mimimum?

x


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Simple Integration

14. a) Use your answer to 9(g) to find a primitive function (indefinite integral) of

g() = sec

2

[Hint: from tables

tan d = ln | sec | +C]b) Use your answer to 9(h) to find a primitive function (indefinite integral) of

h(x) =x sin2x

c) Use your answer to 9(j) to find a primitive function (indefinite integral) off(x) = ln x

15. The curve y = f(x) has dy

dx= 3x2 2x+ 1 and passes through the point (2, 3). Find f(x).

16. Findy where

a)

dy

dx =x+ 1

x for x >0 b) dy

dx = x2+1

x2 for x= 017. Without recourse to tables find

a)

ex dx b)

10

e3x dx

c)

0

sin(2x) dx d)

cos(3x)dx

e)

(2x3 + 3x2 + 4x+ 5)dx f)

1

3x+ 1dx

g) 1

2

1

2x

3

dx

h) For all the above indefinite integrals, check your answers by differentiating.

Logarithms

18. Simplify:

a) log412 log43 b) log216

log28 c) log1/3 729

19. Solve forx:

a) 22x+1

(17)2x + 8 = 0 b) ln x= 3 ln 2 + 2 ln 3 c) log

x

125 =

3

xi


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Answers for Revision Questions

1. Answer for both: the interior and boundary of the triangle with vertices at (0, 0),(2, 0), and (2, 4).

2. a) x 1 b) 1< x


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15. f(x) =x3 x2 +x 3

16. a) y= 23x3/2 + 2

x+C b) y= x 1x + C

17. a)ex +C b) 13(e3 1) c) 0 d) 13 sin(3x) + C e) 12x4 +x3 + 2x2 + 5x+Cf)13 ln|3x+ 1| +C g) ln(1.4)2 =0.168236.

18. a) 1 b) 4/3 c) 6

19. a) 1, 3 b) 72 c) 1/5

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Problems for Chapter 1

Questions marked with [R] are routine, with [H] are harder and with [X] are for MATH1141

only. You should make sure that you can do the easier questions before you tackle the moredifficult questions.

Problems 1.1

1. [R] Express the following sets in words. Graph the sets on the number line (if possible).

a) {x Z : < x < }b) {x R : x2 x 1< 0}c)

{x

Q : x2 = 2

}2. [R] Graph on the number line the following sets.

a) [3, ), (, 3), (, ), (3, 3]b) {x:|x 2|< 5}c) {x: x2 + 4x 5> 0}

3. [R] Sketch the set of points (x, y) which satisfy the following relations.

a) 0y2x and 0x2 b) y/2x2 and 0y4

Problems 1.2, 1.3

4. [R] Solve the following inequalities.

a) x(x 1)> 0 b) (x 1)(x 2)< 0 c) 1x

>12

d) 1

1 x> 1

2 e) x 6

x 1

5. [R] Solve the following inequalities.

a)x+ 13 c) 3x+ 2


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7. [R] True or false:

a) If a > b then 1

a M.

c) Suppose that > 0. Find a real number M (expressed in terms of) such that the

distance between x2 + 1

x2 and its limit is less than whenever x > M.

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6. [R] For each of the following, find the limit off(x) as x tends to infinity and prove fromthe definition that your answer is correct.

a) f(x) = 4x

x+ 7 b) f(x) =

x+ 3

x2

3

c) f(x) =e2x

d) f(x) =sin x

x e) f(x) =

sin3x

x2 + 4

7. [X]

a) With in (0, 1), Sarah solves the inequality|f(x) 4|< and finds that the requiredxvalues satisfy

x

1

,

.

Does limx f(x) exist? Give reasons for your answer.

b) Within (0, 1), Lyndal solves the inequality |g(x)5|< and finds that the inequalityholds for all x satisfying

x

1

,

.

Does limx g(x) exist? Give reasons for your answer.

8. [R] A parcel is dropped from an aeroplane. A simple model, taking into account gravityand air resistance, suggests that the parcels velocity v(t) (in metres per second) is givenbyv(t) = 50(1 et/5), where t is the number of seconds since leaving the plane.

a) Calculate the terminal velocity of the parcel (that is, find limt

v(t)).

b) The parcel never attains its terminal velocity. How long does it take to come within1 metre per second of its terminal velocity?

9. [X] For each question below, give reasons for your answer. [In some cases a single examplewill be sufficient while in other cases a general proof will be required. As a reminder, iff(x) as x then lim

x f(x) does not exist.]

a) If l imx f(x) and limx g(x) do not exist, can limx[f(x) +g(x)] or limx f(x)g(x) exist?

b) If limx f(x) exists and limx[f(x) +g(x)] exists, must limx g(x) exist?

c) If limx f(x) exists and limx g(x) does not exist, can limx[f(x) +g(x)] exist?d) If lim

x f(x) exists and limx f(x)g(x) exists, does it follow that limx g(x) exists?

Problems 2.5

10. [R] Evaluate the following limits.

a) limx3

2x+ 4 b) limx2

x2 4x 2 c) limx1

x3 1x 1 d) limx3

1x 13x 3

11. [R]

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a) Find the left-hand limit limx2

|x 2|x 2.

b) Find the right-hand limit limx

2+

|x 2|x

2

.

c) Does limx2

|x 2|x 2 exist?

12. [R] By finding the left- and right-hand limits first, decide whether or not each of thefollowing limits exist and if so find their values.

a) limx0

x

|x| b) limx2|x2 4|

x 2 c) limx4x 4|x 4| d) limx0

4

x

13. [R]

a) Use the pinching theorem to find limx0

x sin 1

x

.

b) Repeat for limx0

x2 sin 1

2x.

14. [R] Suppose that is a (positive) angle measured in radians and consider the diagrambelow.

O A B

CD

The curve segment CB is the arc of a circle of radius 1 centre O .

a) Write down, in terms of , the length of arc C B and the lengths of the line segmentsCAand DB.

b) By considering areas, deduce that sin tan whenever 0< < 2 .

c) Use the pinching theorem to show that lim0+

sin = 1.

d) Deduce that lim

0

sin

= 1.

15. [H] Discuss the limiting behaviour of cos 1x as x0.

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Problems 3.1, 3.21. [R] Suppose that f : R Ris defined by f(x) =|x|.

a) Show thatfis continuous at 0.

b) Is fcontinuous everywhere? Give brief reasons for your answer.

2. [R] Determine at which points each functionf : R Ris continuous. Give reasons.

a) f(x) = e2x ifx


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Problems 3.4

9. [R] In each case, determine whether or notfattains a maximum on the given interval.Give reasons for your answer.

a) f(x) =x2 4 on [3, 5] b) f(x) =sin(ex) + ln xx2 1

on [2, 4]c) f(x) =x2 4 on (3, 5) d) f(x) =(x2 4) on (3, 5)

10. [H] Suppose that fis a continuous function on R and that limx f(x) = limx

f(x) = 0.

a) Give an example of such a function which has both a maximum value and a minimumvalue.

b) Give an example of such a function which has a minimum value but no maximumvalue.

c) [X] Show that if there is a real number such that f() > 0 then f attains amaximum value on R. [Note that the maximum-minimum theorem only applies to

finiteclosed intervals [a, b].]

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7. [X] The functionf is differentiable at a. Find

limh0

f(a+ph) f(a ph)h

.

8. [R] (An exercise on notation.) Suppose that f(x) =x+ cos 2x. Write down

a) f(x+ 17) b) f(x+n) c) f(2 x2)d) f(2 x2) e) ddxf(2 x2).

Problems 4.4

9. [R] Find dy

dx in terms ofx and y if

a) x3 +y3 =xy b) x2

xy+y2 = 6.

10. [R] Find dy

dx for the curve x4 +y4 = 16. Sketch the graph of the curve.

11. [R] Find the equation of the line tangent to the curve x3 +y3 = 3(x+y) at the point(1, 2).

Problems 4.5

12. [R] Suppose that a and b are real numbers. Find all values ofa and b (if any) such thatthe functions f and g, given by

a) f(x) =

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Problems 4.7

15. [R] At a certain instant the side length of an equilateral triangle isa cm and this lengthis increasing at r cm/sec. How fast is the area increasing?

16. [R] A 5 m ladder is leaning against a vertical wall. Suppose that the bottom of the ladderis being pulled away from the wall at a rate of 1 m/sec. How fast is the area of the triangleunderneath the ladder changing at the instant that the top of the ladder is 4 m from thefloor?

17. [R] A spherical balloon is to be filled with water so that there is a constant increase inthe rate of its surface area of 3 cm2/sec.(The surface area A and volume V of a sphere of radius r is given by A = 4r2 andV = 43r

3.)

a) Find the rate of increase in the radius when the radius is 3 cm.b) Find the volume when the volume is increasing at a rate of 10 cm3/sec.

18. [R]

a) A container in the shape of a right circular cone, of semi-vertical angle tan1(12), isplaced vertex downwards with its axis vertical.

= tan1(12)

Water is poured in at the rate of 10 mm

3

per sec. Find the rate at which the depth,h mm, is increasing when the depth of water in the cone is 50 mm.

b) [H] The cone is filled to a depth of 100 mm and pouring is then stopped. A hole isthen opened at the vertex of the cone and water flows out of the hole at the rateof 50

h mm3 per second, where h is the depth at time t. Show that it takes 200

seconds to empty the cone.

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Problems 5.1

1. [R] Find a real number c which satisfies the conclusions of the mean value theorem foreach function fon the given interval.

a) f(x) =x3 on [1, 2] b) f(x) =

x on [0, 2].

2. [R] Suppose that f(x) = 1/x. Show that there is no real number c in [1, 2] such that

f(c) =f(2) f(1)

2 (1) .

Why does this not contradict the mean value theorem?

3. [R] Consider the functionfgiven by f(x) = (x 2)4 cos(x2 4x + 4). Use the mean valuetheorem to show that f has a zero on the interval [1, 3].

Problems 5.3

4. [R] By using the mean value theorem, show that

a) ln(1 +x)< x whenever x >0;

b) ln(1 x)< x/(1 x) whenever 0< x 0.

5. [R]

a) Use the mean value theorem to show that sin t < t whenever t > 0.

b) Using the pinching theorem and part (a), evaluate the limit limx sin

1

x.

6. [H] Prove that

1 + x

2

1 +x0 for all real numbers x.

b) [X] Suppose that pn(x) =n

k=0

xk

k! whenever n = 1, 2, 3, . . . . Use induction to prove

that

i) if n is even then pn(x)> 0 for all real numbers x, and

ii) ifn is odd then pn(x) has exactly one real root and this root is negative.

13. [R] A wire of length 100 cm is cut into two pieces of lengthx cm and y cm. The piece oflengthx cm is bent into the shape of a square and the piece of length y cm into the shapeof a circle. Find x and y so that the sum of the areas enclosed by the shapes will be

a) a minimum b) a maximum.

14. [X] Suppose that a 0. Find the greatest and least distances from the point (a, 0) tothe ellipse

x2

4 +

y2

1 = 1.

(Have a precise answer before comparing with the given answer.)

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15. [H] Find all the values ofa and x, both in [0, 2], where

f(x) = cos a+ 2 cos(2x) + cos(4x a)

has a horizontal point of inflexion.

Problems 5.8

16. [R] Show that x3 +x 9 = 0 has only one real solution.

17. [R] Suppose that p(x) = x3 12x2 + 45x 51 whenever x R. How many real zerosdoes p have?

Problems 5.9

18. [R]

a) Find a function fthat has the following properties:

f(t) = sin t+t whenever tR,f(0) = 2.

b) Are there any other functions with these properties? Explain your answer.

19. [R] A particle moving along thex-axis has velocity 2t

t2 units per second aftertseconds.Find

a) the distance from the starting point after three seconds;

b) the total distance travelled after three seconds.

Problems 5.10

20. [R] Calculate the following limits.

a) limx

0

ex 1x(3 +x)

b) limx

1

xm 1xn

1

, n= 0 c) limx/2

x /2cos x

d) limx0

ln (1 +x) xx2

e) limx/2

1 sin x1 + cos 2x

f ) lim

x0tan x x

x3

21. [R] Determine the limiting behaviour in the following cases.

a) x3 + 1

x4 + 1 as x b) e

5x

x3 as x

c) e5x

x3 as x d) x sin(1/x) as x

e)

x4 + 1

3

x6 + 1as x

f)

ln(x3 + 1)

ln(x2 + 1)

as x

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22. [H] Find the value of limt0

1

ln(1 +t)+

1

ln(1 t)

.

23. [H] Find (a, b) such that limx0ax

1 +ebx

x2 = 1.

24. [R] Explain why lHopitals Rule cannot be used to find limx

4x+ sin x

2x sin x . Use anothermethod to find this limit.

25. [R] Show that the functionf, given by

f(x) =

e2x ifx02x+ 1 if x


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Problems 6.1

1. [R] Suppose that the functions f : [0, ) [1, ) and g : [1, ) [0, ) are given byf(x) =

1 +x2 and g(x) =

x2 1.

a) By calculating (f g)(x) and (g f)(x), verify that g is the inverse function to f.b) What are the domains off g and g f?

2. [R]

a) Suppose thatf : R R is given by f(x) = 3x+ 1. Findf1(x). Sketch the graphoffand the graph of its inverse function, f1, on the same diagram.

b) The functiong : (, 0] R is defined byg(x) =x2+1. Write down the domain and

range of the inverse function g1

and find a formula for g1

(x). Find the derivativeofg1.

Problems 6.2, 6.3

3. [R] Show that the functionf : R R, given byf(x) =x3+3x+1, has an inverse functionwhose domain is R.

4. [R] Suppose that f : R Ris given by f(x) = 4x+ cos x.a) Show thatfhas an inverse function g .

b) [X] By using the inverse function theorem, find g (2).

5. [R] Suppose that f : R Ris defined by f(x) =x3 3x+ 1.a) Show thatf : R Ris not a one-to-one function.b) Find all p ossible intervalsI ofR, each as large as possible, such that the restricted

function f : I R has an inverse. What is the domain of each of correspondinginverse function?

6. [H]

a) Can you find a quadratic function from R to Rwhich is one-to-one?

b) Can you find a cubic function from Rto Rwhich is not one-to-one?

Problems 6.4

7. [H] For each functionf : R R given below, find all possible intervals I ofR, each aslarge as possible, such that the restricted function f : I R is one-to-one. State therange of each restricted function f : I R. What can you say about existence, domainof definition, continuity and differentiability of the corresponding inverse functions?

a) f(x) =x(x2 1)(x+ 2)b) f(x) = (x+ 1)17

c) f(x) =|x| |x+ 1|24
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Problems 6.4, 6.5

8. [R] Simplify each expression without using a calculator.

a) sin1

(3/2) b) cos(cos1

(2/5)) c) sin1

(sin(5/3))d) cos1(cos(/3)) e) cos(sin1(3/5)) f) sin(tan1(3/5))g) sec1(2) h) sin1(sin x) when 2 x 32

9. [R] Sketch the graph off : [1, 3] R, where f(x) = cos1(x 2).

10. [R] Show that

a) d

dxcos1 x

=

11 x2 b)

d

dxtan1 x

=

1

1 +x2.

11. [R] Differentiate

a) cos1(2x) b) sin1

x c) tan1(2x 3).

12. [R] Prove that sin1 x+ cos1 x is constant. For what values ofxis this valid and whatis the constant?

13. [H] Suppose that f(x) = tan1 x+ tan1(1/x) whenever x= 0.a) Show thatf(x) = 0 whenever x= 0.b) Hence evaluatefon the intervals (0,

) and (

, 0).

c) How do you account for this result geometrically?

14. [H]

a) Draw the graph of cosec x.

b) Show that cosec restricted to the interval (0, 2 ] has an inverse function. Sketch thegraph of the inverse and calculate its derivative.

15. [X]

a) Show that 2 tan1

2 = cos1

(3/5).b) Show that cos1(1 2x2) = 2 sin1 x whenever 0x1.c) Suppose thatq(x) = cos1(1 2x2). Is qdifferentiable at 0?

16. [H] A function f : R Ris defined by

f(x) =

x tan1

1x

ifx >0

ax+b ifx0,

where a and b are real numbers. Find all values ofa and b such that f is differentiable

at 0.

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17. [H] A lighthouse containing a revolving beacon is located 3 km from P, the nearest pointon a straight shoreline. The beacon revolves with a constant rotation rate of 4 revolutionsper minute and throws a spot of light onto the shoreline. How fast is the spot of lightmoving when it is (a) at Pand (b) at a point on the shoreline 2 km from P?

18. [H] A picture 2 metres high is hung on a wall with its bottom edge 6 metres above theeye of the viewer. How far from the wall should the viewer stand for the picture to subtendthe largest possible vertical angle with her eye?

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Problems 7.11. [R] Find the maximal domain and range of the functionf, given byf(x) =

5 + 4x x2,

and sketch its graph.

2. [R] Write down the period of each of the following functionsf(where possible). Determinewhich are odd or even. Sketch the graph of each function.

a) f(x) = sin 3x b) f(x) = 1 + sin(2x/3) c) f(x) =x sin x

d) f(x) = tan 3x e) f(x) = cos2 x f) f(x) = sin x+ cos x

3. [R] Suppose that f is an odd function (not everywhere zero). Determine whether eachfunction g below is odd, even or neither.

a) g(x) =x2f(x) b) g(x) =x3f(x) c) g(x) =x2 +f(x)

d) g(x) =x3 +f(x) e) g(x) = sin(f(x)) f) g(x) =f(cos x)

4. [R] For each function f, identify any vertical and oblique asymptotes and hence sketchthe graph. (Do not use calculus.)

a) f(x) =x+ 2 + 1

x

3

b) f(x) =x2 2

x+ 1 c) [H] f(x) =

x3 7x+ 8x2 +x

6

5. [R] Sketch the following curves, showing their main features.

a) y= x2 + 1

x2 b) y =

x 1x 2 c) y= e

x2/2

d) y = xex e) y2 =x(x 4)2 f) y= x2

x 2g) y=

x2 1x2 2x h) y = x cos

1 x

6. [H] (Longer rather than difficult) Suppose thaty = 3x2 10x+ 33x2 + 10x+ 3

.

a) Find the values ofx for which y0. b) Find the asymptotes.c) Find the turning points. d) Find the domain and range.

e) Sketch the graph.

Problems 7.2

7. [R] Sketch the curves given by the following parametric equations. Also find, where

possible, a Cartesian equation for the curve.

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a) x= 4 cos t, y = 5 sin t

b) x= 3 sec t, y = 2tan t

c) x= t3, y = t2

d) x= et

cos t, y = et

sin t

.

8. [R] For each of the curves given in parametric form by

a)

x= 1 ty = 1 +t

b)

x= 3t+ 2

y = t4 1 c)

x= cos t

y = sin t,

i) find the points on the curve corresponding tot=1, 0, 1, and 2;ii) find any point on the curve wherey = 0;

iii) find dy

dxas a function oft.

9. [R]

a) Find the equation of the normal to the curve x = t

t+ 1, y =

t

t 1 at the point Pwhent = 2.

b) Eliminatet from the above equations and find the gradient of the normal at P usingthe Cartesian form.

10. [X] A curve is given in terms of the parametert by x = t3, y = 3t2.

a) What is the equation of the curve? Can you sketch it?b) Show that the equation of the chord joining the points with parameterst1, t2 is

(t21+t1t2+t22)y = 3(t1+t2)x+ 3t

21t

22.

c) Show that the equation of the tangent att is ty = 2x+t3.

d) Suppose thatPis a point with coordinates (a, b) and that Pdoes not lie on on thecurve or on the y -axis.

i) Show that either one or three tangents may be drawn fromPto the given curve.Illustrate on a sketch the region in which P must lie so that there are threetangents from Pto the curve.

ii) Assume thatPlies in this region and let Q1, Q2, Q3 denote the points of contactof the tangents from P to the curve. Show that the centroid of the triangleQ1Q2Q3 is the point (2a, 2b).

11. [H] Consider a fixed circle of radius 1 centred at the origin and a smaller circle of radius 14initially centred at (34 , 0). The smaller circle rolls (without slipping) around the inside rimof the larger circle such that the centre Q of the smaller circle moves in an anticlockwisedirection. A pointP, fixed on the rim of the smaller circle and initially with coordinates(1, 0), traces out a curve as the smaller circle moves inside the larger circle.

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x

y

O Q P

Initial configuration

x

y

O

Q

P

Configuration after motion has begun

The goal of this question is to find the Cartesian form of the trajectory ofP. Let denotethe angle (in radians) between OQand the positivex-axis, as shown in the above diagram.

a) Explain why

OQ = 34(cos , sin ).

b) [X] Explain why

QP = 14(cos(3), sin(3)).c) Show that

OP= (cos3 , sin3 ).

(You may find techniques from MATH1131 Algebra useful here.)

d) Hence the trajectory ofP is given by

x= cos3 , y= sin3 , 02.

By using an appropriate trigonometric identity, eliminate to find the cartesianequation of the trajectory ofP.

e) Sketch the curve corresponding to this equation. (This curve is called an astroidafterthe Greek word for star.)

Problems 7.3

12. [R] The following points are given in polar coordinate form. Plot them on a diagram andfind their Cartesian coordinates.

a) (3, 0) b) (6, 7/6) c) (2, 7/4)

13. [R] Convert these Cartesian coordinates into polar forms with r0 and < .a) (3, 0) b) (1, 1) c) (2, 23)d) (0, 1) e) (23, 2) f) (23, 2)

14. [R] Sketch the graph corresponding to each polar equation.

a) r= 4 b) = 2 c) r= 3, for 0.

15. [R]

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a) Expressr = 6 sin , where 0, in Cartesian form and hence draw its graph.b) Repeat this forr = 2 cos , where/2/2.

16. [R] Sketch the graph corresponding to each polar equation.

a) r= 2 + sin b) r= 3 + cos c) r= 2 2cos d) r= 2| cos | e) r= 3| sin6| f) r=|tan 2 | ( < < )

17. [H] The hyperbolic spiral is described by the equation r = a whenever > 0, where

a is a positive constant. Using the fact that lim0

sin

= 1, show that the liney = a is a

horizontal asymptote to the spiral. Sketch the spiral.

18. [H] Show thatr = 5

3

2 cos

is the polar equation of an ellipse by finding the Cartesian

equation of the curve (and completing the square).

19. [X]

a) For what values of isr2 = 25 cos2 defined?

b) Sketch the graph of this curve. What difference would it make if you allowed negativevalues ofr?

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Problems 8.1, 8.2

1. [R]

a) By taking the partitionPn =

0,1

n,2

n, . . . , 1

of the interval [0,1], calculate the

lower sumSPn(f) and the upper sum SPn(f) for each function f.i) f(x) = 1

ii) f(x) =x

iii) f(x) =x2

[You may needn

k=1

k2 = 16n(n+ 1)(2n+ 1).]

iv) f(x) =x3

[You may needn

k=1

k3 = 14n2(n+ 1)2.]

v) f(x) =

1 ifxQ0 ifx / Q

b) By taking the limit as n for each sum SPn(f) and SPn(f) calculated in (a),either calculate

10

f(x) dx, or show that fis not Riemann integrable.

2. [R] An electrical signalS(t) has its amplitude |S(t)|tested (sampled) every 110 of a second.It is desired to estimate the energy over a period of half a second, given exactly by 1

2

0|S(t)|2 dt

12

.

The results of the measurement are shown in the following table:

t .1 .2 .3 .4 .5

|S(t)| 60 50 50 45 55e(t) 5 3 7 4 10

a) Using the above data forS(t), set up an appropriate Riemann sum and compute anapproximate value for the energy.

b) It is known that the signal varies by an amount of at moste(t), as shown above, ineach 110 second period. Calculate upper and lower bounds for the energy.

3. [X] Consider the partitionPn of [1, 2], given byPn ={q0, q1, q2, . . . , q n} where qn = 2.(Notice that (i) the divisions are not of equal width and (ii) 1 < q < 2 and q 1 asn .) Iff(x) =xj for some positive integer j, then evaluate the integral 2

1f(x) dx

by calculating the limit limn

SPn

(f) of the corresponding lower Riemann sums.

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Problems 8.3, 8.4

4. [R] Find the area of the region bounded by the liney= x and the parabola y= x2 2.

5. [R] Find

a)

94

x3 xx3/2

dx b)

24

|x| dx.

6. [H] Find a function f which satisfies the integral equation x0

tf(t) dt=

0x

(t2 + 1)f(t) dt+x.

7. [R] Explain why

11

1

x2dx =

1

x

11

=1 1 =2 is not valid.

8. [H]

a) Suppose thatfis a continuous increasing (and hence invertible) function on [a, b]. Ifc= f(a), d = f(b) and a,b,c,d0, then explain why d

cf1(t) dt= bd ac

ba

f(x) dx.

b) Use this to find

11/2

sin1 x dx.

9. [H] Suppose that U(x) =u(x).

a) Find V (x) ifV(x) = (a x)U(x) + x0

U(t) dt where a is a constant.

b) Hence show that

a0

U(x) dx= aU(0) +

a0

(a x)u(x) dx.

Problems 8.5

10. [H] Suppose thatf(t) =tand F(x) = x0

f(t) dt, wheret is the greatest integer lessthan or equal to t. Use a graph offto sketch Fon the interval [1, 3]. Is Fcontinuous?Where is F differentable?

11. [H] Suppose thatf(t) = sin(t2). Sketch the graph offon the interval [0, 3]. Use this to

sketch the graph ofF on the interval [0, 3], where F(x) =

x0

f(t) dt. Indicate whereF

has local maxima and minima.

12. [R] Find F(x) for each function F : R Rgiven below.

a) F(x) =

x0

sin(t2) dt b) F(x) =

x30

sin(t2) dt

c) F(x) = 1

x3sin(t2) dt d) F(x) =

x3

x

sin(t2) dt

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13. [R] Find d

dx

4x

(5 4t)5 dt.

Problems 8.6

14. [R]

a) Suppose thatf(x) = 1

x. By considering the lower Riemann sum for f with respect

to the partition n

n,n+ 1

n ,

n+ 2

n , . . . ,

2n

n

of [1, 2], show that

ln 2 = limn

1

n+ 1+

1

n+ 2+ + 1

2n .b) Suppose thatf(x) =

11 x2 .

i) Show thatfis increasing on the interval [0, 12 ].

ii) Find the upper Riemann sum for fwith respect to the partition 0

2n,

1

2n,

2

2n,

3

2n, . . . ,

n

2n

of [0, 12 ].

iii) Hence evaluate

limn

14n2 12 +

14n2 22 +

14n2 32 + +

14n2 n2

.

Problems 8.7

15. [R] Evaluate the following integrals by inspection.

a)

x ex

2

dx b)

sin

x

x dx

c) 1

02x(1 +x2)3 dx d)

a

a

x2a3 x3 dx (a >0)e)

/20

cos3 x sin x dx f) [H]

01

t2 +t4 dt

Problems 8.8

16. [R] Use a substitution to evaluate the following integrals.

a)

dx

1 +

x b)

x(5x 1)19 dx

c) 1 x(1 +x)3

dx d) 4

0

dx

5 + x36


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17. [X] Use the substitution u = t t1 to find

1 +t2

1 +t4dt.

Problems 8.9

18. [R] Use integration by parts to evaluate the following integrals.

a)

10

x e5x dx b)

x2 cos x dx c)

ln x dx

d)

0.50

sin1 x dx e) e1

x7 ln x dx f)

0

x2 cos 2x dx

g)

ex cos x dx h)

tan1 x dx i) [H]

/40

sec3 d

Problems 8.10

19. [R] Evaluate the following improper integrals or show that they diverge.

a)

0

e5xdx b) 1

e0.01xdx c)0

dx

4 +x2

d)

x3 ex4

dx e)

2

dx

(x 1)3/2 f)e

dx

x ln x

20. [H] Prove that

0

xnexdx= n! whenever n = 0, 1, 2, . . . .

21. [H]

a) Find limR

RR

x

1 +x2dx.

b) Find limR

2RR

x

1 +x2dx.

c) Does

x

1 +x2dx converge? Explain.

Problems 8.11

22. [R] Use the inequality form of the comparison test to determine whether or not thefollowing improper integrals converge.

a)

1

11 +x4

dx b)

2

13

x2 x dx c)2

1

ln xdx

23. [R] Use the limit form of the comparison test to determine whether or not the followingimproper integrals converge.

a)

2

x

2x3

1

dx b)

1

2x 1x2 + 2

dx c)

2

1x6

1dx

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24. [R] Use a comparison test to determine whether or not the following improper integralsconverge.

a)

1

3x+ sin x+ 2

2x3 x+ 8 dx b)

4

4x3 x+ 5x4 x2 + 1

dx c) [H]

2

ln t

t3/2dt

25. [H] Find all real numberss such that the improper integral1

xs

1 +xdx

is convergent.

26. [H] Find all real numbersp such that

2

1

x(ln x)pdx converges.

27. [H] For which pairs of numbers (a, b) does the improper integral1

xb

(1 +x2)a dxcon-

verge?

Problems 8.12

28. [R] Given a positive real number x, let (x) denote the number of primes less than orequal to x. The function Li with domain (1, ) is given by

Li(x) =

x

2

1

ln tdt

and is known as the logarithmic integral function. It has the property that

Li(x)

(x)1

when x is sufficiently large.

a) Evaluate(10), (20) and (3.14159).

b) Suppose thatx >0. What does (x)

x represent?

c) Find ddxLi(x) and Li(2).

d) By applying the mean value theorem to Li on the interval [2, 106], find a lower boundfor Li(106).

e) If x is large then(x)

x (x)

x

Li(x)

(x) =

Li(x)

x .

Using this approximation and your answer to part (d), find an approximate lower

bound for (106)

106 .

Note: There are 78, 498 primes less than one million so the actual value of (106)

106 is

0.078498.

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29. [R] The function erf : R Ris defined by the formula

erf(x) = 2

x0

et2

dt.

The function erf is an error function and can be used to calculate the probability that ameasurement has an error in a given range of values.

a) Calculate erf(x).

b) Explain why erf is an increasing function onR.

c) [H] Show that erf is an odd function.

d) i) By calculating Riemann sums with respect to the partition{0, 14 , 12 , 34 , 1}, findupper and lower bounds for erf(1).

ii) Explain why et2 < et whenever t > 1.

iii) Hence show that1 e

t2

dtconverges and find an upper bound for this improperintegral.

iv) Using your answers to (i) and (iii), find an upper bound for limx erf(x). (In fact,

limx erf(x) = 1 but this is not so easy to prove.)

e) Sketch the graph of erf.

f) Explain why erf has an inverse function erf1 and sketch its graph.

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Problems 9.1, 9.2

1. [R]

a) Write down the definition of ln x, where x >0.

b) Explain why d

dxln x=

1

x whenever x >0.

c) Suppose thatr is a rational number and that x and y are positive real numbers.

i) By first differentiating ln(xy) with respect tox, show that ln(xy) = ln x + ln y.

ii) Use the same technique to show that

ln

xy

= ln x ln y and ln(xr) =r ln x.

2. [R]

a) Prove, using upper and lower Riemann sums and the definition of ln x, that ln 2 0 b) lim

x0+xa ln x, a >0

c) limx0+ xx

d) limx0+ x2/ ln x

e) limx x

1/x f) limx a

1/x, a >0

g) limx

1 +

a

x

xh) lim

x x100 ex

i) limx p(x) e

x , where p is any polynomial.

10. [H] Prove that the functionsf : (1, )Rand g : (1, ) R, given by

f(x) = ln(1 +x)

x x2

2 and g(x) = x

x2

2

+x3

3 ln(1 +x),

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are increasing on (0, ). Deduce that

x x2

2 0.

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Problems 10.1, 10.2

1. [R] Define sinh x and cosh x. Hence show that

a) d

dx(cosh 6x) = 6 sinh6x;

b) ln(sinh x)< x ln 2 wheneverx >0.

2. [R] By expressing the following hyperbolic functions in terms of sinh xand cosh x, findthe derivative of each function f given below.

a) f(x) = tanh x b) f(x) = sech x c) f(x) = coth x

3. [R] In each case, find f(x).

a) f(x) = sinh(3x2) b) f(x) = cosh( 1x) c) f(x) = sinh(ln x)

Problems 10.3

4. [R]

a) Given the formula sinh(A + B) = sinh A cosh B+ cosh A sinh B, find a formula forsinh2x. By differentiation or otherwise, find a formula for cosh 2x.

b) [H] Using the results of part (a), express sinh 3x as a cubic polynomial in sinh x.

Hence, or otherwise, find

sinh3 x dx.

5. [R] Show that cosh x+sinh x= ex. Deduce that (cosh x+sinh x)n = cosh nx+sinh nx.

6. [H] Consider the hyperbola x2 y2 = 1, wherex1.

A(t)

x

y

(cosh t, sinh t)

0 1

a) Using the definitions of cosh and sinh, prove that, for every real number t, the

point (cosh t, sinh t) lies on the hyperbola.

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b) When t >0, let A(t) denote the shaded region in the diagram. Explain why

A(t) = 1

2cosh t sinh t

cosh t1

x2 1 dx.

c) By first calculatingA(t), prove that A(t) = t2

.

Problems 10.4

7. [R] Evaluate the following integrals.

a)

cosh(4x) dx b)

13ln 2

0sinh3x dx

c)

cosh2 x dx d)

sinh(

x)

x dx

Problems 10.5

8. [R] Simplify cosh(sinh1(3/4)), cosh1(cosh(3)) and sinh(tanh1(5/13)).

9. [R] Show that

a) d

dx

cosh1 x

=

1x2 1 b)

d

dx

tanh1 x

=

1

1 x2 .

10. [R] Show that

a) cosh1

x= ln(x+

x2

1) x[1, )b) tanh1 x=

1

2ln

1 +x

1 x

x(1, 1).

11. [R] Find dy

dx if

a) y= sinh1(2x)b) y= tanh1(1/x)c) y= cosh1(sec x) whenever 0 < x < /2.

Problems 10.6

12. [R] Find

a)

dx

1 + 4x2 b)

1/20

dx

1 x2 c)

dxx2 + 4x+ 13

.

13. [X] Sketch the function sech1. What is its maximal domain? For y= sech1x, showthat

a) dy

dx=

1x

1 x2 b) y= ln

1 +

1 x2x

.

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Answers to selected problems

Chapter 1

1. a) The set of integers between and .c) The empty set.

3. Answer for both: the interior and boundary of the triangle with vertices at (0, 0), (2, 0) and(2, 4).

4. a) x 1 b) 1< x < 2 c) x 0d) 1< x


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18. a) If p(x) =a0+a1x+ + anxn thenp(q(x)) =a0+a1q(x) +a2(q(x))2 + +an(q(x))n.Products and sums of polynomials are again polynomials.

b) Yes.

Chapter 2

1. a) 1 b) 2 c) 0

d) Doesnt exist ( ). e) 5 f) Doesnt exist.

2. a) 0 b) 0

4. b) 0

5. a) 1 b) M= 10 (best possible) c) M= 1/

will do.

6. a) 4 b) 0 c) 0 d) 0 e) 0

7. a) Not necessarily, as the information given indicates only that the inequality holds for asubset of (1, ).

b) Yes. In fact one can prove that limx

g(x) = 5 from the definition of the limit by taking

M to be 1

.

8. a) 50 metres per second b) 5 l n 5019.56 seconds after leaving the plane.

9. a) Yes. If limit off(x) as x does not exist and f(x)= 0, thenlimx

(f(x) f(x)) = 0 and limx

(f(x)/f(x)) = 1.

b) Yes, sinceg (x) = (f(x) +g(x)) f(x).c) No, as in (b).

d) No. For example iff(x) = 0 for allx and limx

g(x) does not exist, we have

limx

(f(x)g(x)) = 0.

10. a) 10 b) 4 c) 3 d) 1/9

11. a) 1 b) 1 c) No

12. a) Doesnt exist. b) Doesnt exist. c) Doesnt exist. d) Doesnt exist.

13. a) 0 b) 0

14. a) |CB |= , |CA|= sin , |DB|= tan .

15. Neither the left-hand nor right-hand limits exist due to wild oscillatory behaviour.

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Chapter 3

1. b) Yes

2. a) Continuous everywhere. b) Continuous everywhere except at/2.

3. k= 8

5. Use the intermediate value theorem.

9. a) Yes b) Yes c) No d) Yes

Chapter 4

2. a) 5(4x3 + 21x6) b) (4x32)(4x2+2x+4)+(x42x)(8x+2)c) (16y y4)/(y3 + 8)2 d) (2x2 4)/(x2 4)1/2e) 4/(t2 4)3/2 f ) 3 cos 3y+ 12 cos 2y sin2yg) (4x3 x4)ex h) x ln(x3 + 1) + 3x2(x2 + 1)/2(x3 + 1)i) sec2 x j) tan x

3. a) 0 b) 0 c) f(0) = 0

4. a) i)x= 0 ii) all x b) i) all x ii) all xc) i) x

=

2 ii) x

=

2

7. 2pf(a)

8. a) x+ 17+ cos 2x b) 1 2 sin 2x c) 2x2+cos2(2x2)d) 1 2 sin2(2 x2) e) 2x(1 2 sin 2(2 x2))

9. a) dy

dx=

3x2 yx 3y2 b)

dy

dx = (y 4xxy)/(4yxy x)

11. y= 2

12. a) (i)b= 0 (ii)a= 1,b = 0

b) (i)b= 1 (ii)a= 2,b = 1.

13. a= 1, b= 0

14. a) f(8.01) f(8) = 2b) i) y= (x 8)/12 + 2

ii) f(8.01) (8.01 8)/12 + 2 = 2 + 11200c) The approximation in (b) is much better.

15.

3 ar/2

16. 7/8

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23. (a, b) = (2, 2) or (2, 2)

26. a) 1/2

27. a=1/2,b = 128. c) a= b = 0

Chapter 6

2. a) f1(x) =1

3(x 1)

b) g1(x) =x 1, Dom(g1) = [1, ),Range(g1) = (, 0], (g1)(x) = 1

2x1

4. b) 1/3

5. b) The restriction off to (, 1] has an inverse with domain (, 3],the restriction off to [1, 1] has an inverse with domain [1, 3], andthe restriction off to [1, ) has an inverse with domain [1, ).

6. a) No b) Yes

7. a) The graph is symmetric about x =12 , which surely gives a local maximum of f(x).

There will be four (maximal) intervals where fwill have an inverse. Try this exercise onMaple. The commands plot, diffand solveshould suffice.

b) fis one-to-one; f1(x) =x1/17 1 is not differentiable when x = 0.c) Ican be one of four intervals.

8. a) /3 b) 2/5 c) /3 d) /3e) 4/5 f) 3/

34 g) /3 h) x

11. a) 2/1 4x2 b) 1/(2x x2 ) c) 2/(4x2 12x+ 10)

12. Differentiate;1x1; /2.

13. b) f(x) =/2 when x >0 and f(x) =/2 when x 1.

16. a= /2, b = 0

17. a) 24 km/min b) 104/3 km/min

18.

48 metres

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Chapter 7

1. [1, 5], [0, 3], upper half of circle.

2. a) period 2/3, odd b) period 3, neither

c) not periodic, even d) period /3, odd

e) period , even f) 2

3. odd, even, neither, odd, odd, even.

4. The asymptotes are

a) x= 3, y= x+ 2 b) x=1, y= x 1 c) x=3, x= 2, y= x 1.

6. a)x3,13

< x 13

b) x =13

, x=3, y= 1 c) (1, 14

), (1, 4) d) Domain:x= 3,

1

3

, Range: (

,

4], [

1

4

,

).

7. a) x2

16+

y2

25 = 1, ellipse b)

x2

9 y

2

4= 1, hyperbola

c) y = x2/3 d) spiral

8. a) ii) (2, 0) iii)1b) ii) (5, 0), (1, 0) iii) 4t3/3c) ii) (1, 0), (1, 0) iii) cot t

9. a) 3x 27y+ 52 = 0 b) 19

10. a)y = 3x

2

3

.

11. b) Hint: the length of one particular arc of the larger circle equals the length of one arc onthe smaller circle.

d) x2/3 +y2/3 = 1

12. a) (3, 0) b) (33,3) c) (2, 2)

13. a) (3, ) b) (

2, 3/4) c) (4, 2/3)d) (1, /2) e) (4, 5/6) f) (4, 5/6)

14. a) Circle, centre (0,0), radius 4

b) A ray in the second quadrant

c) A spiral of Archimedes

15. a) Circle, centre (0,3), radius 3

b) Circle, centre (1,0), radius 1

16. The following sketches are a guide to shape only.

a)

x

y

b)

x

y

c)

x

y

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d)

x

y

e)

x

y

f)

x

y

18. (x 2)2

9 +

y2

5 = 1

Chapter 8

1. a) i) SPn

(f) =SPn

(f) = 1

ii) SPn

(f) = 121 1n , SPn(f) = 12 1 + 1n

iii) SPn(f) = 16

1 1n

2 1n

, SPn(f) =

16

1 + 1n

2 + 1n

v) SPn(f) = 1, SPn(f) = 0

b) i) 1 ii) 12

iii) 13

iv) 14

(v) Not Riemann integrable

2. a)

1365 = 36.95

b)

1690.9 = 41.12 and the lower bound is

1078.9 = 32.85

4. 4.5

5. a) 82.4 b) 10

6. f(x) = 1

x2 +x+ 1

7. 1x is not differentiable on all of [1, 1] so the FTC doesnt apply.

8. a) Draw a picture! b) 5/12 3/2

10. F is continuous everywhere, but not differentiable at the integers.

12. a) sin x2 b) 3x2 sin x6 c) 3x2 sin x6 d) 3x2 sin x6 sin x2

13. (5 4x)5

14. biii) 6

.

15. a) 12 ex2 +C b) 2cos x+C c) 15/4

d) 4

2 a9/2

9 e) 1/4 f) (2

2 1)/3

16. a) 2

x 2 ln(1 + x) +C b) 125 121(5x 1)21 + 120(5x 1)20

+C

c) x/(x+ 1)2 +C d) 4 10 ln(7/5)

17. 1

2tan1 t

2 12 t for t= 0

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18. a) 4e5+125

b) x2 sin x+ 2x cos x 2 sin x+Cc) x(ln(x) 1) +C d) 12+

32 1

e) 7e8+164

f) 2

g) ex

2(cos x+ sin x) h) x tan1

x ln 1 +x2

+Ci)

22

+ 12

ln(1 +

2)

19. a) 1/5 b) diverges c) /4

d) 0 e) 2 f) diverges

21. a) 0 b) ln 2 c) No

22. a) convergent b) divergent c) divergent

23. a) convergent b) divergent c) convergent

24. a) convergent b) divergent c) convergent

25. s < 0

26. p >1

27. The integral converges whenever 2a b >1.

28. a) 4, 8, 2

c) Li(x) = 1lnx >0 so Li is an increasing function; Li(2) = 0.

d) Li(106

) 106

2

6ln10 .e) (10

6)x 0.07238.

29. a) 2

ex2

d) (i) 0.749< erf(1)< 0.928 (iii) 1/e (iv) 1.344

Chapter 9

2. a) A partition into 7 equal parts will suffice

3. a) 3x2/2(x3 + 1) b) ex for x >0, ex forx 61/5)

4. a) 12

ln(1 +e2x) b) e1/xc) 3x/ ln 3 d) e

x

4

e) (lnx)2

2 f ) ln | sin x|

7. b) e

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8. a) 3x ln 3 b)

x3 3x2 + 1

1/5 3x2

5(x3 3) 2x

5(1 +x2)

c) (sinx)sinx cos x (1+ln(sin x)) d) cos(xsinx) xsin x

cos x ln x+sin x

x

9. a) 0 b) 0 c) 1 d) e2 e) 1

f) 1 g) ea h) 0 i) 0

Chapter 10

2. a) sech2 x b) sechx tanh x c) cosech2x

3. a) 6x cosh(3x2) b) sinh(1/x)x2 c)

1

2+ 1

2x2

4. a) sinh 2x= 2 cosh x sinh x ; cosh 2x= cosh2 x+ sinh2 x

b) 14

(13

cosh3x 3 cosh x) or 13

cosh3 x cosh x

7. a) sinh4x4 b) 112 c) (2x+ sinh 2x)/4 d) 2 cosh

x

8. 5/4, 3, 5/12

11.

12. a) 2/

1 + 4x2 b) 11x2 for|x|> 1 c) sec x

13. a) 12

sinh1 2x b) tanh1 12

= 12

ln 3 c) sinh1x+23
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