bosquejo de funciones con gdc numeros y algebra estudios matematicos

1. Consider the function f(x) = x3 + x

48, x ≠ 0.

(a) Calculate f(2).(2)

(b) Sketch the graph of the function y = f(x) for –5≤ x ≤ 5 and –200 ≤ y ≤ 200.(4)

(c) Find f′(x).(3)

(d) Find f′(2).(2)

(e) Write down the coordinates of the local maximum point on the graph of f.(2)

(f) Find the range of f.(3)

(g) Find the gradient of the tangent to the graph of f at x = 1.(2)

There is a second point on the graph of f at which the tangent is parallel to the tangent at x = 1.

(h) Find the x-coordinate of this point.(2)

(Total 20 marks)

2. The function f(x) is defined by f(x) = 1.5x + 4 + x

6, x ≠ 0.

(a) Write down the equation of the vertical asymptote.(2)

IB Questionbank Mathematical Studies 3rd edition 1

(b) Find f′(x).(3)

(c) Find the gradient of the graph of the function at x = –1.(2)

(d) Using your answer to part (c), decide whether the function f(x) is increasing or decreasing at x = –1. Justify your answer.

(2)

(e) Sketch the graph of f(x) for –10 ≤ x ≤ 10 and –20 ≤ y ≤ 20.(4)

P1 is the local maximum point and P2 is the local minimum point on the graph of f(x).

(f) Using your graphic display calculator, write down the coordinates of

(i) P1;

(ii) P2.(4)

(g) Using your sketch from (e), determine the range of the function f(x) for –10 ≤ x ≤ 10.(3)

(Total 20 marks)

3. The function f(x) = 5 – 3(2–x) is defined for x ≥ 0.

(a) (i) On the axes below sketch the graph of f(x) and show the behaviour of the curve as x increases.


(ii) Write down the coordinates of any intercepts with the axes.

(4)

(b) Draw the line y = 5 on your sketch.(1)

(c) Write down the number of solutions to the equation f(x) = 5.(1)

(Total 6 marks)


4. Consider the function y = 3 cos(2x) + 1.

(a) Sketch the graph of this function for 0 ≤ x ≤ 180°.

(3)

(b) Write down the period of the function.(1)

(c) Using your graphic display calculator find the smallest possible value of x,0 ≤ x ≤ 180°, for which 3 cos(2x) + 1 = 2.

(2)(Total 6 marks)


5. The diagram shows a sketch of the function f(x) = 4x3 – 9x2 – 12x + 3.

diagram not to scale

(a) Write down the values of x where the graph of f(x) intersects the x-axis.(3)

(b) Write down f′(x).(3)

(c) Find the value of the local maximum of y = f(x).(4)

Let P be the point where the graph of f(x) intersects the y-axis.

(d) Write down the coordinates of P.(1)

(e) Find the gradient of the curve at P.(2)


The line, L, is the tangent to the graph of f(x) at P.

(f) Find the equation of L in the form y = mx + c.(2)

There is a second point, Q, on the curve at which the tangent to f(x) is parallel to L.

(g) Write down the gradient of the tangent at Q.(1)

(h) Calculate the x-coordinate of Q.(3)

(Total 19 marks)

6. The graph of the function f(x) = 2 cos (4x) – 1, where 0° ≤ x ≤ 90°, is shown on the diagram below.

(a) On the diagram draw the graph of the function g(x) = sin(2x) – 2,for 0° ≤ x ≤ 90°.

(2)

(b) Write down the number of solutions to the equation f(x) = g(x), for 0° ≤ x ≤ 90°.(1)


(c) Write down one value of x for which f(x) > g(x), for 0° ≤ x ≤ 90°.(1)

f(x) < g(x) in the interval a < x < b.

(d) Use your graphic display calculator to find the value of

(i) a;

(ii) b.(2)

(Total 6 marks)

7. Consider the function f(x) = x3 – 3x2 – 24x + 30.

(a) Write down f(0).(1)

(b) Find f′(x).(3)

(c) Find the gradient of the graph of f(x) at the point where x = 1.(2)

The graph of f(x) has a local maximum point, M, and a local minimum point, N.

(d) (i) Use f′(x) to find the x-coordinate of M and of N.

(ii) Hence or otherwise write down the coordinates of M and of N.(5)

(e) Sketch the graph of f(x) for –5 ≤ x ≤ 7 and –60 ≤ y ≤ 60. Mark clearly M and N on your graph.

(4)


Lines L1 and L2 are parallel, and they are tangents to the graph of f(x) at points A and B respectively. L1 has equation y = 21x + 111.

(f) (i) Find the x-coordinate of A and of B.

(ii) Find the y-coordinate of B.(6)

(Total 21 marks)

8. Consider the function f(x) = 3x + 2

12

x, x ≠ 0.

(a) Differentiate f(x) with respect to x.(3)

(b) Calculate f′(x) when x = 1.(2)

(c) Use your answer to part (b) to decide whether the function, f, is increasing or decreasing at x = 1. Justify your answer.

(2)

(d) Solve the equation f′(x) = 0.(3)

(e) The graph of f has a local minimum at point P. Let T be the tangent to the graphof f at P.

(i) Write down the coordinates of P.

(ii) Write down the gradient of T.

(iii) Write down the equation of T.(5)

(f) Sketch the graph of the function f, for –3 ≤ x ≤ 6 and –7 ≤ y ≤ 15. Indicate clearly the point P and any intercepts of the curve with the axes.

(4)


(g) (i) On your graph draw and label the tangent T.

(ii) T intersects the graph of f at a second point. Write down the x-coordinate of this point of intersection.

(3)(Total 22 marks)

9. (a) Sketch the graph of y = 2x for –2 ≤ x ≤ 3. Indicate clearly where the curve intersects the y-axis.

(3)

(b) Write down the equation of the asymptote of the graph of y = 2x.(2)

(c) On the same axes sketch the graph of y = 3 + 2x – x2. Indicate clearly where this curve intersects the x and y axes.

(3)

(d) Using your graphic display calculator, solve the equation 3 + 2x – x2 = 2x.(2)

(e) Write down the maximum value of the function f(x) = 3 + 2x – x2.(1)

(f) Use Differential Calculus to verify that your answer to (e) is correct.(5)

(Total 16 marks)


10. (a) Sketch the graph of the function f (x) = 4

32

x

x, for −10 x 10. Indicating clearly the

axis intercepts and any asymptotes.(6)

(b) Write down the equation of the vertical asymptote.(2)

(c) On the same diagram sketch the graph of g (x) = x + 0.5.(2)

(d) Using your graphical display calculator write down the coordinates of one of the points of intersection on the graphs of f and g, giving your answer correct to five decimal places.

(3)

(e) Write down the gradient of the line g (x) = x + 0.5.(1)

(f) The line L passes through the point with coordinates (−2, −3) and is perpendicular to the line g (x). Find the equation of L.

(3)(Total 17 marks)

11. Consider the curve y = x3 2

2

3x – 6x – 2

(a) (i) Write down the value of y when x is 2.

(ii) Write down the coordinates of the point where the curve intercepts the y-axis.(3)

(b) Sketch the curve for –4 ≤ x ≤ 3 and –10 ≤ y ≤ 10. Indicate clearly the information found in (a).

(4)

(c) Find x

y

d

d.

(3)


(d) Let L1 be the tangent to the curve at x = 2.Let L2 be a tangent to the curve, parallel to L1.

(i) Show that the gradient of L1 is 12.

(ii) Find the x-coordinate of the point at which L2 and the curve meet.

(iii) Sketch and label L1 and L2 on the diagram drawn in (b).(8)

(e) It is known that x

y

d

d > 0 for x < –2 and x > b where b is positive.

(i) Using your graphic display calculator, or otherwise, find the value of b.

(ii) Describe the behaviour of the curve in the interval –2 < x < b.

(iii) Write down the equation of the tangent to the curve at x = –2.(5)

(Total 23 marks)


12. The following curves are sketches of the graphs of the functions given below, but in a different order. Using your graphic display calculator, match the equations to the curves, writing your answers in the table below.

diagrams not to scale

A B C

D E F

0

0

0

0 0

0

y

x

y

x

y

x

y

x

y

x

y

x

Function Graph label

(i) y = x3 + 1

(ii) y = x2 + 3

(iii) y = 4 − x2

(iv) y = 2x + 1

(v) y = 3−x + 1

(vi) y = 8x − 2x2 − x3

(Total 6 marks)


13. (a) Sketch the graph of the function y =1+ 2

)2(sin x for 0 x 360 on the

axes below.(4)

9 0 1 8 0 2 7 0 3 6 0

2

1

– 1

– 2

x

y

(b) Write down the period of the function.(1)

(c) Write down the amplitude of the function.(1)

(Total 6 marks)

14. It is not necessary to use graph paper for this question.

(a) Sketch the curve of the function f (x) = x3 − 2x2 + x − 3 for values of x from −2 to 4, giving the intercepts with both axes.

(3)

(b) On the same diagram, sketch the line y = 7 − 2x and find the coordinates of the point of intersection of the line with the curve.

(3)

(c) Find the value of the gradient of the curve where x =1.7.(2)

(Total 8 marks)


15. The following figures show the graphs of y = f (x) with f (x) chosen to be various cubic functions. The value of a is positive.

A B C

D E

y

x

a

a

a

y

xa

a

a

y

x

y y

aa

aa

xx

(a) In the table below, write the letter corresponding to the graph of y = f (x) in the space next to the cubic function.

(Note: one of the graphs is not represented in this table)

cubic function f (x) graph label

f (x) = x3 + a

f (x) = (x – a)3 + a

f (x) = x3

f (x) = (x – a)3

(b) State which one of the graphs represents a function that has a positive gradient for every value of x.

(c) State how many of the graphs have the x-axis as a tangent at some point.(Total 6 marks)


16. Two functions f (x) and g (x) are given by

f (x) = 1

12 x

,

g (x) = x , x 0.

(a) Sketch the graphs of f (x) and g (x) together on the same diagram using values of x between –3 and 3, and values of y between 0 and 2. You must label each curve.

(b) State how many solutions exist for the equation 1

12 x

– x = 0.

(c) Find a solution of the equation given in part (b).(Total 6 marks)

17. (a) Sketch the graph of y = 3 + 1

3

x for −10 x 10.


(b) Write down the equations of

(i) the horizontal asymptote;

(ii) the vertical asymptote.(Total 6 marks)

18. (a) Sketch the graph of the function y = 2x2 – 6x + 5.

(b) Write down the coordinates of the local maximum or minimum of the function.

(c) Find the equation of the axis of symmetry of the function.(Total 6 marks)

19. (a) Sketch a graph of y = x

x

2 for –10 ≤ x ≤ 10.

(b) Hence write down the equations of the horizontal and vertical asymptotes.(Total 6 marks)

20. (a) Sketch the graph of the function f : x 1 + 2 sin x, where x , –360° ≤ x ≤ 360°.(4)

(b) Write down the range of this function for the given domain.(2)

(c) Write down the amplitude of this function.(1)

(d) On the same diagram sketch the graph of the function g : x sin 2x, where x ,–360° ≤ x ≤ 360°.

(4)


(e) Write down the period of this function.(1)

(f) Use the sketch graphs drawn to find the number of solutions to the equationf (x) = g (x) in the given domain.

(1)

(g) Hence solve the equation 1 + 2 sin x = sin 2x for 0° ≤ x ≤ 360°.(4)

(Total 17 marks)

21. (a) On the same graph sketch the curves y = x2 and y = 3 – x

1 for values of x from 0 to 4 and

values of y from 0 to 4. Show your scales on your axes.(4)

(b) Find the points of intersection of these two curves.(4)

(c) (i) Find the gradient of the curve y = 3 – x

1 in terms of x.

(ii) Find the value of this gradient at the point (1, 2).(4)

(d) Find the equation of the tangent to the curve y = 3 – x

1 at the point (1, 2).

(3)(Total 15 marks)


22. The functions f and g are defined by

f : x x

x 4, x , x 0

g : x x, x

(a) Sketch the graph of f for –10 ≤ x ≤ 10.(4)

(b) Write down the equations of the horizontal and vertical asymptotes of the function f.(4)

(c) Sketch the graph of g on the same axes.(2)

(d) Hence, or otherwise, find the solutions of x

x 4 = x.

(4)

(e) Write down the range of function f.(2)

(Total 16 marks)


23. The figure below shows the graphs of the functions y = x2 and y = 2x for values of x between –2 and 5.

The points of intersection of the two curves are labelled as B, C and D.

y

x

C

A

B

D

– 2 – 1 1 2 3 4 5

2 0

1 5

1 0

5

(a) Write down the coordinates of the point A.(2)

(b) Write down the coordinates of the points B and C.(2)

(c) Find the x-coordinate of the point D.(1)

(d) Write down, using interval notation, all values of x for which 2x ≤ x2.(3)

(Total 8 marks)


24. At the circus a clown is swinging from an elastic rope. A student decides to investigate the

motion of the clown. The results can be shown on the graph of the function f (x) = (0.8x)(5 sin 100x), where x is the horizontal distance in metres.

(a) Sketch the graph of f (x) for 0 ≤ x ≤ 10 and –3 ≤ f (x) ≤ 5.(5)

(b) Find the coordinates of the first local maximum point.(2)

(c) Find the coordinates of one point where the curve cuts the y-axis.(1)

Another clown is fired from a cannon. The clown passes through the points given in the table below:

Horizontal distance (x) Vertical distance (y)

0.00341 0.0102

0.0238 0.0714

0.563 1.69

1.92 5.76

3.40 10.2

(d) Find the correlation coefficient, r, and comment on the value for r.(3)

(e) Write down the equation of the regression line of y on x.(2)

(f) Sketch this line on the graph of f (x) in part (a).(1)

(g) Find the coordinates of one of the points where this line cuts the curve.(2)

(Total 16 marks)


bosquejo de funciones con gdc numeros y algebra estudios matematicos

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