IB SL calculus exam practice

1. Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5.

Find the area of the region enclosed by the graphs of f and g.(Total 6 marks)

2. The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) direction.

diagram not to scale

The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water level. After t seconds, the height of the bucket above the water level is given by h = a sin bt + 2.

(a) Show that a = 4.(2)

The wheel turns at a rate of one rotation every 30 seconds.

(b) Show that b = 15

π

.(2)

In the first rotation, there are two values of t when the bucket is descending at a rate of

0.5 m s–1.

(c) Find these values of t.(6)

IB Questionbank Maths SL 1

(d) Determine whether the bucket is underwater at the second value of t.(4)

(Total 14 marks)

3. A gradient function is given by 5e10

d

d 2 x

x

y

. When x = 0, y = 8. Find the value of y whenx = 1.

(Total 8 marks)

4. Let f(x) = x cos x, for 0 ≤ x ≤ 6.

(a) Find f′(x).(3)


(b) On the grid below, sketch the graph of y = f′(x).

(4)(Total 7 marks)


5. Let f(x) = ex sin 2x + 10, for 0 ≤ x ≤ 4. Part of the graph of f is given below.

There is an x-intercept at the point A, a local maximum point at M, where x = p and a local minimum point at N, where x = q.

(a) Write down the x-coordinate of A.(1)

(b) Find the value of

(i) p;

(ii) q.(2)

(c) Find

q

pxxf d)(

. Explain why this is not the area of the shaded region.(3)

(Total 6 marks)


6. Consider f(x) = x ln(4 – x2), for –2 < x < 2. The graph of f is given below.

(a) Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the x-axis.

(i) Find the x-coordinate of P and of Q.

(ii) Consider f(x) = k. Write down all values of k for which there are exactly two solutions.

(5)

Let g(x) = x3 ln(4 – x2), for –2 < x < 2.

(b) Show that g′(x) = )4ln(3

4

2 222

4

xxx

x

.(4)

(c) Sketch the graph of g′.(2)

(d) Consider g′(x) = w. Write down all values of w for which there are exactly two solutions.

(3)(Total 14 marks)


7. The velocity v m s–1 of an object after t seconds is given by v(t) = tt 315 , for 0 ≤ t ≤ 25.

(a) On the grid below, sketch the graph of v, clearly indicating the maximum point.

(3)

Let d be the distance travelled in the first nine seconds.

(b) (i) Write down an expression for d.

(ii) Hence, write down the value of d.(4)

(Total 7 marks)

8. Let f′(x) = –24x3 + 9x2 + 3x + 1.

(a) There are two points of inflexion on the graph of f. Write down the x-coordinates of these points.

(3)

(b) Let g(x) = f″(x). Explain why the graph of g has no points of inflexion.(2)

(Total 5 marks)


9. Let f(x) = x ln(4 – x2), for –2 < x < 2. The graph of f is shown below.

The graph of f crosses the x-axis at x = a, x = 0 and x = b.

(a) Find the value of a and of b.(3)

The graph of f has a maximum value when x = c.

(b) Find the value of c.(2)

(c) The region under the graph of f from x = 0 to x = c is rotated 360° about the x-axis.Find the volume of the solid formed.

(3)

(d) Let R be the region enclosed by the curve, the x-axis and the line x = c, between x = a and x = c.

Find the area of R.(4)

(Total 12 marks)


10. Let f(x) = x(x – 5)2, for 0 ≤ x ≤ 6. The following diagram shows the graph of f.

Let R be the region enclosed by the x-axis and the curve of f.

(a) Find the area of R.(3)

(b) Find the volume of the solid formed when R is rotated through 360° about the x-axis.

(4)


(c) The diagram below shows a part of the graph of a quadratic function g(x) = x(a – x). The graph of g crosses the x-axis when x = a.

The area of the shaded region is equal to the area of R. Find the value of a.(7)

(Total 14 marks)

11. Let f(x) = 3sinx + 4 cos x, for –2π ≤ x ≤ 2π.

(a) Sketch the graph of f.(3)

(b) Write down

(i) the amplitude;

(ii) the period;

(iii) the x-intercept that lies between 2

π and 0.

(3)

(c) Hence write f(x) in the form p sin (qx + r).(3)


(d) Write down one value of x such that f′(x) = 0.(2)

(e) Write down the two values of k for which the equation f(x) = k has exactly two solutions.

(2)

(f) Let g(x) = ln(x + 1), for 0 ≤ x ≤ π. There is a value of x, between 0 and 1, for whichthe gradient of f is equal to the gradient of g. Find this value of x.

(5)(Total 18 marks)

12. Let f(x) = x

4

πcos5

and g(x) = –0.5x2 + 5x – 8, for 0 ≤ x ≤ 9.

(a) On the same diagram, sketch the graphs of f and g.(3)

(b) Consider the graph of f. Write down

(i) the x-intercept that lies between x = 0 and x =3;

(ii) the period;

(iii) the amplitude.(4)

(c) Consider the graph of g. Write down

(i) the two x-intercepts;

(ii) the equation of the axis of symmetry.(3)

(d) Let R be the region enclosed by the graphs of f and g. Find the area of R.(5)

(Total 15 marks)


13. Let f (x) = ex (1 – x2).

(a) Show that f ′ (x) = ex (1 – 2x – x2).(3)

Part of the graph of y = f (x), for – 6 x 2, is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively.

(b) Write down the equation of the horizontal asymptote.(1)

(c) Write down the value of r and of s.(4)

(d) Let L be the normal to the curve of f at P(0, 1). Show that L has equation x + y = 1.

(4)

(e) Let R be the region enclosed by the curve y = f (x) and the line L.

(i) Find an expression for the area of R.

(ii) Calculate the area of R.(5)

(Total 17 marks)


14. Let f(x) = x cos (x – sin x), 0 ≤ x ≤ 3.

(a) Sketch the graph of f on the following set of axes.

(3)

(b) The graph of f intersects the x-axis when x = a, a ≠ 0. Write down the value of a.(1)

(c) The graph of f is revolved 360° about the x-axis from x = 0 to x = a.Find the volume of the solid formed.

(4)(Total 8 marks)

15. Let f(x) = e2x cos x, –1 ≤ x ≤ 2.

(a) Show that f′(x) = e2x (2 cos x – sin x).(3)

Let the line L be the normal to the curve of f at x = 0.

(b) Find the equation of L.(5)


The graph of f and the line L intersect at the point (0, 1) and at a second point P.

(c) (i) Find the x-coordinate of P.

(ii) Find the area of the region enclosed by the graph of f and the line L.(6)

(Total 14 marks)

16. The function f is defined as f (x) = ex sin x, where x is in radians. Part of the curve of f isshown below.

There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects the x-axis at the point C.

(a) Write down the x-coordinate of the point C.(1)

(b) (i) Find f (x).

(ii) Write down the value of f (x) at the point B.(4)

(c) Show that f (x) = 2ex cos x.(2)

(d) (i) Write down the value of f (x) at A, the point of inflexion.

(ii) Hence, calculate the coordinates of A.(4)


(e) Let R be the region enclosed by the curve and the x-axis, between the origin and C.

(i) Write down an expression for the area of R.

(ii) Find the area of R.(4)

(Total 15 marks)

17. The function f (x) is defined as f (x) = 3 + 52

1

x , x 2

5

.

(a) Sketch the curve of f for −5 x 5, showing the asymptotes.(3)

(b) Using your sketch, write down

(i) the equation of each asymptote;

(ii) the value of the x-intercept;

(iii) the value of the y-intercept.(4)

(c) The region enclosed by the curve of f, the x-axis, and the lines x = 3 and x = a, is revolved through 360 about the x-axis. Let V be the volume of the solid formed.

(i) Find

252

1

52

69

xxdx.

(ii) Hence, given that V =

3ln3

3

28π

, find the value of a.(10)

(Total 17 marks)


18. Let f (x) = 22

3

qx

xp

, where p, q +.

Part of the graph of f, including the asymptotes, is shown below.

(a) The equations of the asymptotes are x =1, x = −1, y = 2. Write down the value of

(i) p;

(ii) q.(2)

(b) Let R be the region bounded by the graph of f, the x-axis, and the y-axis.

(i) Find the negative x-intercept of f.

(ii) Hence find the volume obtained when R is revolved through 360 about thex-axis.

(7)

(c) (i) Show that f ′ (x) =

22

2

1

13

x

x

.

(ii) Hence, show that there are no maximum or minimum points on the graph off.

(8)


(d) Let g (x) = f ′ (x). Let A be the area of the region enclosed by the graph of g and the x-axis, between x = 0 and x = a, where a > 0. Given that A = 2, find the value of a.

(7)(Total 24 marks)

19. Consider the function f (x) e(2x–1) +

12

5

x, x 2

1

.

(a) Sketch the curve of f for −2 x 2, including any asymptotes.(3)

(b) (i) Write down the equation of the vertical asymptote of f.

(ii) Write down which one of the following expressions does not represent an area between the curve of f and the x-axis.

2

1 f (x)dx

2

0 f (x)dx

(iii) Justify your answer.(3)

(c) The region between the curve and the x-axis between x = 1 and x = 1.5 is rotatedthrough 360 about the x-axis. Let V be the volume formed.

(i) Write down an expression to represent V.

(ii) Hence write down the value of V.(4)

(d) Find f ′ (x).(4)


(e) (i) Write down the value of x at the minimum point on the curve of f.

(ii) The equation f (x) = k has no solutions for p k < q. Write down the value of p and of q.

(3)(Total 17 marks)

20. The function f is defined as f (x) = (2x +1) e−x, 0 x 3. The point P(0, 1) lies on the graph of f (x), and there is a maximum point at Q.

(a) Sketch the graph of y = f (x), labelling the points P and Q.(3)

(b) (i) Show that f ′ (x) = (1− 2x) e−x.

(ii) Find the exact coordinates of Q.(7)

(c) The equation f (x) = k, where k , has two solutions. Write down the range of values of k.

(2)

(d) Given that f (x) = e−x (−3 + 2x), show that the curve of f has only one point of inflexion.

(2)

(e) Let R be the point on the curve of f with x-coordinate 3. Find the area of the region enclosed by the curve and the line (PR).

(7)(Total 21 marks)


21. Let h (x) = (x – 2) sin (x – 1) for –5 x 5. The curve of h (x) is shown below. There is a minimum point at R and a maximum point at S. The curve intersects the x-axis at the points (a, 0) (1, 0) (2, 0) and (b, 0).

– 5 – 4 – 3 – 2 – 1 1 2 3 4 5 x

y

S( , 0 )b( , 0 )a

R

4

3

2

1

– 1

– 2

– 3

– 4

– 5

– 6

– 7

(a) Find the exact value of

(i) a;

(ii) b.(2)

The regions between the curve and the x-axis are shaded for a x 2 as shown.

(b) (i) Write down an expression which represents the total area of the shaded regions.

(ii) Calculate this total area.(5)

(c) (i) The y-coordinate of R is –0.240. Find the y-coordinate of S.

(ii) Hence or otherwise, find the range of values of k for which the equation(x – 2) sin (x – 1) = k has four distinct solutions.

(4)(Total 11 marks)


22. Consider the function f (x) = cos x + sin x.

(a) (i) Show that f (– 4

π

) = 0.

(ii) Find in terms of p, the smallest positive value of x which satisfies f (x) = 0.(3)

The diagram shows the graph of y = ex (cos x + sin x), – 2 x 3. The graph has a maximum turning point at C(a, b) and a point of inflexion at D.

6

4

2

1 2 3– 2 – 1

D

C ( a , b )

y

x

(b) Find x

y

d

d

.(3)

(c) Find the exact value of a and of b.(4)

(d) Show that at D, y = 4

π

e2 .(5)

(e) Find the area of the shaded region.


(2)(Total 17 marks)

23. Let the function f be defined by f (x) = 31

2

x , x –1.

(a) (i) Write down the equation of the vertical asymptote of the graph of f .

(ii) Write down the equation of the horizontal asymptote of the graph of f .

(iii) Sketch the graph of f in the domain –3 x 3.(4)

(b) (i) Using the fact that f (x) = 23

2

)1(

6–

x

x

, show that the second derivative

f (x) =

33

3

)1(

1–212

x

xx

.

(ii) Find the x-coordinates of the points of inflexion of the graph of f .(6)

(c) The table below gives some values of f (x) and 2f (x).

x f (x) 2f (x)

1 1 2

1.4 0.534188 1.068376

1.8 0.292740 0.585480

2.2 0.171703 0.343407

2.6 0.107666 0.215332

3 0.071429 0.142857

(i) Use the trapezium rule with five sub-intervals to approximate the integral

.d3

1 xxf

(ii) Given that

3

1dxxf

= 0.637599, use a diagram to explain why your answeris greater than this.

(5)(Total 15 marks)


24. The diagram below shows a sketch of the graph of the function y = sin (ex) where –1 x 2, and x is in radians. The graph cuts the y-axis at A, and the x-axis at C and D. It has a maximum point at B.

– 1 0 1 2

A

B

C D

y

x

(a) Find the coordinates of A.(2)

(b) The coordinates of C may be written as (ln k, 0). Find the exact value of k.(2)

(c) (i) Write down the y-coordinate of B.

(ii) Find x

y

d

d

.

(iii) Hence, show that at B, x = ln 2

π

.(6)

(d) (i) Write down the integral which represents the shaded area.

(ii) Evaluate this integral.(5)

(e) (i) Copy the above diagram into your answer booklet. (There is no need to

copy the shading.) On your diagram, sketch the graph of y = x3.

(ii) The two graphs intersect at the point P. Find the x-coordinate of P.(3)

(Total 18 marks)


25. Note: Radians are used throughout this question.

(a) Draw the graph of y = p + x cos x, 0 x 5, on millimetre square graph paper, using a scale of 2 cm per unit. Make clear

(i) the integer values of x and y on each axis;

(ii) the approximate positions of the x-intercepts and the turning points.(5)

(b) Without the use of a calculator, show that p is a solution of the equationp + x cos x = 0.

(3)

(c) Find another solution of the equation p + x cos x = 0 for 0 x 5, giving your answer to six significant figures.

(2)

(d) Let R be the region enclosed by the graph and the axes for 0 x p. Shade R onyour diagram, and write down an integral which represents the area of R .

(2)

(e) Evaluate the integral in part (d) to an accuracy of six significant figures. (If you consider it necessary, you can make use of the result

.)cos)cossin(d

dxxxxx

x

(3)(Total 15 marks)


26. The function f is given by

21

2–1)(

x

xxf

(a) (i) To display the graph of y = f (x) for –10 x 10, a suitable interval for y, a y b must be chosen. Suggest appropriate values for a and b .

(ii) Give the equation of the asymptote of the graph.(3)

(b) Show that .

)1(

2–2)(

22

2

x

xxf

(4)

(c) Use your answer to part (b) to find the coordinates of the maximum point of the graph.

(3)

(d) (i) Either by inspection or by using an appropriate substitution, find

dxxf )(

(ii) Hence find the exact area of the region enclosed by the graph of f, the x-axis and the y-axis.

(8)(Total 18 marks)


27. In this question you should note that radians are used throughout.

(a) (i) Sketch the graph of y = x2 cos x, for 0 x 2 making clear the approximatepositions of the positive x-intercept, the maximum point and the end-points.

(ii) Write down the approximate coordinates of the positive x-intercept, the maximum point and the end-points.

(7)

(b) Find the exact value of the positive x-intercept for 0 x 2.(2)

Let R be the region in the first quadrant enclosed by the graph and the x-axis.

(c) (i) Shade R on your diagram.

(ii) Write down an integral which represents the area of R.(3)

(d) Evaluate the integral in part (c)(ii), either by using a graphic display calculator, or by using the following information.

xdd

(x2 sin x + 2x cos x – 2 sin x) = x2 cos x.(3)

(Total 15 marks)


28. In this part of the question, radians are used throughout.

The function f is given by

f (x) = (sin x)2 cos x.

The following diagram shows part of the graph of y = f (x).

A

C

B

y

xO

The point A is a maximum point, the point B lies on the x-axis, and the point C is a pointof inflexion.

(a) Give the period of f.(1)

(b) From consideration of the graph of y = f (x), find to an accuracy of one significant figure the range of f.

(1)


(c) (i) Find f (x).

(ii) Hence show that at the point A, cos x = 31

.

(iii) Find the exact maximum value.(9)

(d) Find the exact value of the x-coordinate at the point B.(1)

(e) (i) Find )(xfdx.

(ii) Find the area of the shaded region in the diagram.(4)

(f) Given that f (x) = 9(cos x)3 – 7 cos x, find the x-coordinate at the point C.(4)

(Total 20 marks)


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IB SL calculus exam practice