Review For Unit 3 Chapter 3 Part 2

*** sin2x is the same as (sin x)2

*** KNOW THE UNIT CIRCLE!!

1) sin2x + cos2x =

2) tan2x + 1 =

3) 1 + cot2x =

4) sin(a + b) =

5) cos(a + b) =

Inverse Trig Functions

We must restrict the domain of sine, cosine, & tangent in order for their inverses to be functions

y = sin x

D:

*restrict domain to

arcsineR:

y = cos x

D:

* restrict domain to

arccosineR:

y = tan x

D:

*restrict domain to

arctangentR:

Practice:

Derivative Rules

1. Power Rule

2. Products

3. Quotients

4. Trig Derivatives

5. Chain Rule

6. Implicit Differentiation

7. Inverse Trig Derivatives

8. Exponential Derivatives

9. Derivatives of logarithms

10. Logarithmic

Differentiation

Trig Derivative IntroductionNAME

Match the six trig functions on the right with their correct derivative on the left. Do this by paying attention to where slope is positive, negative, or 0. If slope is positive, y will have positive y-values (ABOVE the axis). If slope is negative, y will have negative y-values (BELOW the axis). If slope is 0, the y graph will hit the axis there.

1. y = sin(x)

A)

2. y =cos(x)

B)

3. y = tan(x)

C)

4. y =cot(x)

D)

5. y = sec(x)

E)

6. y = csc(x)

F)

Practice with Trig and the Chain Rule

Find for the following problems:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Proof of the rule for the derivative of y = sin x

Given

Equivalent to given statement

Derivative of both sides with respect to x.

Simplifyimplicit differentiation

Solve for dy/dx

Pythagorean Identity--

Easier understood notation

Substitute--

Implicit Differentiation & Inverse TrigName___________________

Find by implicit differentiation.

1.

EMBED Equation.DSMT4

2.

3.

4.

5.

6.

Use implicit differentiation to find the slope-intercept equation of the tangent line at the indicated point.

1)

EMBED Equation.DSMT4

2)

Implicit Differentiation & Inverse TrigName___________________Use implicit differentiation to find the slope-intercept equation of the normal line at the indicated point.

1)

2)

Find the derivative of the function. Simplify where possible.

1)

2)

3)

4)

5)

Implicit Differentiation & Inverse Trig Derivatives Practice Name ___________________ 1. Differentiate:

_____ 2. If , find .

_____ 3. Determine the slope of the line normal to the graph of: at

_____ 4. Given that find the equation of the tangent line at (1,0).

_____ 5. Differentiate .

Implicit Differentiation and Inverse Trig Derivatives Practice

_____ 6. Differentiate: .

_____ 7. Find if

_____ 8. Find

_____ 9. Find the equation of the tangent line to at x = 3.

_____ 10. If the position of a particle is given by , find the acceleration of the particle after 2 seconds?

FRQ #1 (no calculator)

Name ___________________________Consider the curve given by .

(a) Show that .

(b) Find all points on the curve whose x-coordinate is 1, and write an equation for the tangent line at each of these points.

(c) Find the x-coordinate of each point on the curve where the tangent line is vertical.

(a)

(b)

(c)

FRQ #2 (no calculator)

Name _________________________________The horizontal position, in centimeters, of a particle is given by the function , where t is in seconds.

a) Find the velocity function, , and use it to find the velocity at time .

b) In the interval 0 to seconds, when is the particle at rest?

c) Is the particle moving left, right, or stationary at time ?

d) Find the acceleration function, a(t). What is the particles acceleration at time ?

a)

b)

c)

d)

FRQ #3 (no calculator)

Name _______________________Let be the function given by .

(a) Find the domain of .

(b) Describe the symmetry, if any, of the graph of .

(c) Find . Reduce completely.

(d) Find the slope of the line normal to the graph at x = 5.

(a)

(b)

(c)

(d)

Calculus: Unit 3 Sudoku

Name_________________________

Clues:

A.If , = _________

B.The slope of the curve at (0, 0) is _______

C.Determine if .

D.If the position of a particle is represented by ,

determine the velocity when the acceleration is 0.

E.Evaluate with

F.Determine the slope of the tangent line to at (0, 0).

G.What is the slope of the tangent line to at (3, -3)?

H.If , determine .

I.The surface area of a snowball rolling down a hill is increasing at a rate of

. What is the radius of the snowball at the moment when the radius

is increasing at ?

Complete the problems on the next page. Enter the numbers into the puzzle corresponding to answers of the lettered problems. Then, complete the Sudoku puzzle using the following:

You must fill each row, column, and 3 EMBED Equation.3 3 box with the numbers 1 to 9 such that:

Each number can appear only once in each row.

Each number can appear only once in each column.

Each number can appear only once in each 3 EMBED Equation.3 3 box.

There is only one solution for this puzzle.

There is only one solution for this puzzle.

B

3

1

8

9

H

7

6

C

4

E

3

4

7

8

3

7

1

4

2

I

4

8

9

A

G

7

5

1

9

F

D

1

6

8

PAGE 6

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