ib math hl: cumulative review 01irecognizethelionbyhispaw.com/ib/2012_hl.pdf · ib math hl:...

Name: ________________________

IB Math HL: Cumulative review 01

This assignment is to be done at your leisure over the summer. We designed it to help you see how IB exams assess

the following concepts (some of the concepts were covered in Geometry or Algebra II, most were covered in

advanced pre-calculus and AP calculus):

1. Sequences and series

2. Counting principles

3. Binomial theorem

4. Function basics

5. Quadratic functions

6. Graphs of functions

a. Using the GDC

b. Transformations of functions

c. Asymptotic behavior of rational functions

d. Graphs of trigonometric functions

e. Curve shape / optimization (calculus)

7. Factor and remainder theorems

8. Logarithmic and exponential functions

9. Matrices

10. Circular geometry

11. Solutions of triangles

12. Trig equations and identities

13. Derivative skills

14. Integration skills

15. Motion

16. Related rates

17. Area and volume

18. Separable differential equations

The purpose of this assignment We know that you want to be successful on the IB exams. The purpose of this packet is to support you in achieving

this goal. Hence, you should understand that this packet is for you, not for us. While the packet is required, it

might not be turned in or graded.

Due date This packet is due on the first day of school: August 18, 2010.

Graphics Display Calculators (GDCs) and sig figs Each problem indicates whether a GDC may or may not be used. If you use a GDC to find a decimal approximation,

be sure to give your answer correct to three significant figures – no more and no less.

Solutions and extra copies Solutions and extra copies of this problem set are posted on Mr. Peck’s webpage:

http://www.bvsd.org/schools/centaurus/Teachers/FredPeck

Some final notes All of these problems come from past IB exams. At an IB test pace, this packet should take about six hours to

complete. Don’t worry if it takes longer – the important thing is to review and refresh. If something is not

familiar to you, look it up!!!

Math takes space, and IB does not give credit for answers with no work to support them. Please show all of

your work ON SEPARATE PAPER.

If you have questions, please feel free to contact me at: [email protected]

http://www.bvsd.org/schools/centaurus/Teachers/FredPeck

mailto:[email protected]

Key topic #1: Sequences and series

Objectives

Students will be able to:

1. Explain the difference between sequences and series (Larson: section 8.1)

2. Recognize arithmetic and geometric sequences and series (Larson: sections 8.1, 8.2,& 8.3)

3. Apply the formulae for the general terms of arithmetic sequences and series in problem-solving situations (Larson: section 8.2)

4. Apply the formulae for the general terms of geometric sequences and series in problem-solving situations (Larson: section 8.3)

5. Recognize situations in which an infinite geometric series has a finite value, and find the sum of such infinite geometric series. (Larson: section 8.3)

6. Model real-world phenomena with arithmetic and geometric sequences and series (Larson: sections 8.2

and 8.3)

1. No GDC allowed

The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find

the first term, a, and the common difference, d, of the sequence.

(Total 4 marks)

2. No GDC allowed

Find the sum to infinity of the geometric series

(Total 3 marks)

3. No GDC allowed

The sum of an infinite geometric sequence is , and the sum of the first three terms is 13.

Find the first term.

(Total 3 marks)

....3

16–812–

2113

Key topic #2: Counting principles

Objectives


1. In grouping situations, distinguish between permutations and combinations (Larson: section 8.6)

2. Calculate:

a. with a GDC (Larson: section 8.6)

b. by hand and with a GDC (Larson: section 8.6)

3. Find the number of permutations or combinations of size that can be created from a group of

size . (Larson: section 8.6)

4. Find the probability of a particular permutation or combination or size given a group of size

. (Larson: sections 8.6 & 8.7)

4. GDC required

A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find the

number of ways the committee may be chosen.

(Total 6 marks)

5. GDC required

In how many ways can six different coins be divided between two students so that each student receives at least one

coin?

(Total 3 marks)

Key topic #3: Binomial theorem

Objectives


1. Apply the binomial theorem to expand binomials (Larson: section 8.5)

2. Use the binomial theorem to find any given term in a binomial expansion (Larson: section 8.5)

6. GDC required

(a) Find the expansion of (2 + x)5, giving your answer in ascending powers of x.

(b) By letting x = 0.01 or otherwise, find the exact value of 2.015.

(Total 6 marks)

7. GDC required

Find the coefficient of x7 in the expansion of (2 + 3x)

10, giving your answer as a whole number.

(Total 3 marks)

Key topic #4: Function basics

Objectives

1. Students will find the composition of two functions (Alg. II; Larson: section 1.5)

2. Students will find the domain and range of functions (Alg. II; Larson: sections 1.2 & 1.3)

3. Students will find the inverse of functions (Alg. II; Larson: section 1.6)

8. No GDC allowed

The functions f (x) and g (x) are given by f (x) = and g (x) = x2 + x. The function (f ° g)(x) is defined for x ,

except for the interval ] a, b [.

(a) Calculate the value of a and of b.

(b) Find the range of f ° g.

(Total 6 marks)

9. No GDC allowed

Given functions f : x x + 1 and g : x x3, find the function (f ° g)

–l.

(Total 3 marks)

2–x

Key topic #5: Quadratic functions

Objectives

1. Students will express and interpret quadratic functions in three forms: (Alg. II; Larson: section 2.1 and appendix B.3)

a. Standard form: 0ax by c

b. Factored form: 1 2 0a x x x x

c. Vertex form: 0a x h k

2. Students will solve quadratic equations (Alg. II; Larson: section 2.1 and appendix B.3)

3. Students will use the discriminant ( ) to determine the number of solutions to a

quadratic equation (Alg. II)

10. GDC required

Given f (x) = x2 + x(2 – k) + k

2, find the range of values of k for which f (x) > 0 for all real values of x.

(Total 4 marks)

11. GDC required

The equation kx2 – 3x + (k + 2) = 0 has two distinct real roots. Find the set of possible values of k.

(Total 3 marks)

Key topic #6: Graphs of functions

Objectives

Using the GDC


1. Graph arbitrary functions in arbitrary viewing windows using GDCs (Alg. II; Larson: appendix B.3)

2. Find roots (zeros) of functions using their GDCs (Alg. II; Larson: appendix B.3)

3. Find the intersection of two graphs using GDCs (Alg. II; Larson: appendix B.3)

4. Solve inequalities using GDCs (Alg. II; Larson: appendix B.3)

Transformations of functions


5. Transform functions by translating, reflecting, and dilating, both horizontally and vertically

(Alg. II; Larson section 1.4)

6. Explain the effect of the following transformations on the graph of y f x : (Alg. II; Larson section

1.4)

a. ( )y f x b. ( )y f x

c. ( )y f x b d. ( )y f x a

e. ( )y pf x f. ( )y f kx

Asymptotic behavior of functions


7. Find horizontal and vertical asymptotes of rational functions, and explain their reasoning (Larson:

section 2.6)

Graphs of trigonometric functions


8. Draw and interpret graphs of , , and , including:

a. Interpreting - and -intercepts (Larson: sections 4.5 & 4.6)

b. Interpreting asymptotes (Larson: sections 4.5 & 4.6)

c. Interpreting periodic behavior (Larson: sections 4.5 & 4.6)

d. Transformations of the form, (Larson: sections 4.5 & 4.6)

Curve shape / optimization (calculus)

Given or ’ (analytically or graphically), students will be able to:

9. Find the gradient of the graph of at a given value of

10. Write equations for tangent lines

11. Write equations for normal lines

12. Find minima and maxima; solve optimization problems

13. Find intervals on which a function is increasing and decreasing

14. Find points of inflection

15. Determine the concavity of a function

16. Sketch graphs of given and vice-versa

12. GDC required

Given that x > 0, find the solution of the following system of equations:

xy – y = x2 +

(Total 3 marks)

13. GDC required

Solve the inequality x2 – 4 + < 0.

(Total 6 marks)

14. No GDC allowed

The diagram shows a sketch of part of the graph of f (x) = x2 and a sketch of part of the graph of

g (x) = –x2 + 6x – 13

(a) Write down the coordinates of the maximum point of y = g (x).

The graph of y = g (x) can be obtained from the graph of y = f (x) by first reflecting the graph of y = f (x), then

translating the graph of y = f (x).

(b) Describe fully each of these transformations, which together map the graph of y = f (x) onto the

graph of y = g (x).

(Total 3 marks)

38 3

yx

49

x

3

y

x

y f x= ( )

y=g x( )

15. No GDC allowed

The diagram below shows the graph of y1 = f (x). The x-axis is a tangent to f (x) at x = m and

f (x) crosses the x-axis at x = n.

On the same diagram sketch the graph of y2 = f (x – k), where 0 < k < n – m and indicate the coordinates of the points of

intersection of y2 with the x-axis.

(Total 4 marks)

16. No GDC allowed

Find the equations of all the asymptotes of the graph of y = .

(Total 6 marks)

17. No GDC allowed

The graph below represents y = a sin (x + b) + c, where a, b, and c are constants.

Find values for a, b and c. (Total 6 marks)

18. GDC required

Consider the tangent to the curve y = x3 + 4x

2 + x – 6.

(a) Find the equation of this tangent at the point where x = –1.

(b) Find the coordinates of the point where this tangent meets the curve again.

(Total 3 marks)

x

y

mn

y f x = ( )

0

1

45–

4–5–2

2

xx

xx

–2 – 2x

y

4

3

2

1

–1

–2

–3

–4

0

34 , 2

19. No GDC allowed

Let f be the function defined for x > by f (x) = ln (3x + 1).

(a) Find f ′(x).

(b) Find the equation of the normal to the curve y = f (x) at the point where x = 2.

Give your answer in the form y = ax + b where a, b . (Total 6 marks)

20. No GDC allowed

Let f : x esin x

.

(a) Find f (x).

There is a point of inflexion on the graph of f, for 0 < x < 1.

(b) Write down, but do not solve, an equation in terms of x, that would allow you to find the value of x

at this point of inflexion.

(Total 3 marks)

21. No GDC allowed

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

Indicate, and label clearly, on the graph

(a) the points where y = f (x) has minimum points;

(b) the points where y = f (x) has maximum points;

(c) the points where y = f (x) has points of inflexion.

(Total 3 marks)

22. No GDC allowed

A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on the curve y = sin x,

where 0 x n.

(a) Write down an expression for the area of the rectangle.

(b) Find the maximum area of the rectangle.

(Total 3 marks)

3

1

y

x

y f’ x = ( )

Key topic #7: Factor and remainder theorems

Objective

Students will be able to evaluate polynomials at given the remainder when the polynomial is

divided by (Larson section 2.3)

23. No GDC allowed

When the function f (x) = 6x4 + 11x

3 – 22x

2 + ax + 6 is divided by (x + 1) the remainder is –20. Find the value of a.

(Total 4 marks)

24. No GDC allowed

The polynomial f (x) = x3 + 3x

2 +ax + b leaves the same remainder when divided by (x – 2) as when divided by (x +1).

Find the value of a. (Total 6 marks)

Key topic #8: Logarithmic and exponential functions

Objectives

1. Students will model real-world behavior using logarithmic and exponential functions (Larsen: sections 3.1,

3.2, & 3.5)

2. Students will solve logarithmic and exponential equations (Larsen: section 3.4)

3. Students will simplify logarithmic expressions using properties of logarithms. (Larsen: section 3.3)

25. No GDC allowed

Solve the equation 9 log5 x = 25 logx 5, expressing your answers in the form , where p, q .

(Total 6 marks)

26. No GDC allowed

Solve log16 .

(Total 6 marks)

q

p

5

2

1–100

3 2 x

Key topic #9: Matrices

Objectives


1. Perform basic operations with matrices:

o Addition, subtraction, scalar multiplication (Larsen: section 7.5)

o Matrix multiplication (Including properties of matrix multiplication) (Larsen: section 7.5)

o Matrix algebra

2. Calculate the determinant of 2x2 and 3x3 matrices by hand, and interpret the value of the determinant (Larsen: section 7.7)

3. Calculate the inverse of 2x2 matrices by hand, and of 3x3 matrices using a GDC, and use the inverse matrix in problem-solving situations (Larsen: section 7.6)

27. No GDC allowed

Find the values of the real number k for which the determinant of the matrix is equal to zero.

(Total 3 marks)

28. GDC required

The matrix A is given by

A =

Find the values of k for which A is singular.

(Total 6 marks)

29. No GDC allowed

The matrices A, B, C and X are all non-singular 3 × 3 matrices.

Given that A–l

XB = C, express X in terms of the other matrices.

(Total 6 marks)

1

3

2

4

k

k

243

1–1

12

k

k

Key topic #10: Circular geometry

Objectives


1. Measure and interpret angles in degrees and radians (Larsen: section 4.1)

2. Find the length of an arc of a circle. (Geometry; Larsen: section 4.1)

3. Find the area of a sector of a circle (Geometry; Larsen: section 4.1: problems #104-107)

30. GDC required

The diagram below shows a circle centre O and radius OA = 5 cm. The angle = 135°.

Find the area of the shaded region.

(Total 6 marks)

BOA

O

A

B

Key topic #11: Solutions of triangles

Objectives


1. Solve for all missing sides and angles in a triangle when given SSS, SAS, & ASA, using:

a. SOHCAHTOA (Larsen: section 4.3)

b. The sine rule (Law of sines) (Larsen: section 6.1)

c. The cosine rule (Law of cosines) (Larsen: section 6.2)

2. Recognize ambiguous triangles, and solve for both possible values of the missing sides and angles in ambiguous triangles. (Larsen: section 6.1)

3. Find the area of triangles given SAS (Larsen: section 6.1)

31. GDC required

In the triangle ABC, = 30°, BC = 3 and AB = 5. Find the two possible values of .

(Total 6 marks)

32. GDC required

A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the angle between these two

sides is 60°.

(a) Calculate the length of the third side of the field. (3)

(b) Find the area of the field in the form p , where p is an integer.

(3)

Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts by

constructing a straight fence [AD] of length x metres.

(c) (i) Show that the area of the smaller part is given by and find an expression for the area of

the larger part.

(ii) Hence, find the value of x in the form q , where q is an integer.

(8)

(d) Prove that .

(6)

(Total 20 marks)

A B

3

4

65x

3

8

5

DC

BD

Key topic #12: Trig equations and identities

Objectives


1. Simplify trigonometric expressions using the following identities:

a. Pythagorean identities (Larsen: section 5.1)

b. Double angle identities (Larsen: section 5.5)

c. Compound angle identities (sum and difference formulas) (Larsen: section 5.4)

2. Solve trigonometric equations, including those which are quadratic in or (i.e., ) (Larsen: sections 5.3 & 5.4)

33. No GDC allowed

Prove that = tan , for 0 < < , and .

(Total 5 marks)

34. GDC required

The function f is defined on the domain [0, ] by f ( ) = 4 cos + 3 sin .

(a) Express f ( ) in the form R cos ( – ) where 0 < < .

(b) Hence, or otherwise, write down the value of for which f ( ) takes its maximum value.

(Total 6 marks)

35. No GDC allowed

Solve 2 sin x = tan x, where x

(Total 3 marks)

4cos–12cos

2cos–14sin

2

π

4

π

2

π

2–

.2

Key topic #13: Derivative skills

Objectives


1. Differentiate the following “Building block” functions:

a. Power functions ( )

b. All six trig functions

c. and

d. and

2. Differentiate combinations of building-block functions:

a. Addition and subtraction

b. Multiplication

c. Division

d. Composition

3. Differentiate curves where is implicitly related to

36. No GDC allowed

Consider the function y = tan x – 8 sin x.

(a) Find

(b) Find the value of cos x for which

(Total 3 marks)

37. No GDC allowed

For the function f : x x2 1n x, x > 0, find the function f, the derivative of f with respect to x.

(Total 3 marks)

38. GDC required

Let y = e3x

sin (x).

(a) Find .

(b) Find the smallest positive value of x for which = 0.

(Total 6 marks)

39. No GDC allowed

If 2x2 – 3y

2 = 2, find the two values of when x = 5.

(Total 4 marks)

.d

d

x

y

.0d

d

x

y

x

y

d

d

x

y

d

d

x

y

d

d

Key topic #14: Integration skills

Objectives


1. Evaluate the following indefinite integrals:

a. Power functions:

b. Simple trig:

i.

ii.

c. Exponential functions:

d.

e. Inside function times derivative:

f. Integration by substitution

g. Integration by parts

h. Simplification using trig identities

2. Find particular solutions to the above integrals given an initial condition

3. Evaluate definite integrals involving the above patterns.

40. No GDC allowed

Let f (t) =

(Total 3 marks)

41. No GDC allowed

Find the real number k > 1 for which dx = .

(Total 4 marks)

42. No GDC allowed

Given that = ex – 2x and y = 3 when x = 0, find an expression for y in terms of x.

(Total 6 marks)

43. No GDC allowed

The function f ′ is given by f ′(x) = 2sin .

(a) Write down f ″(x).

(b) Given that f = 1, find f (x).

(Total 6 marks)

.d)(Find.

2

1–1

3

53

1

ttf

t

t

k

x1

211

23

x

y

d

d

25

πx

2

π

Key topic #15: Motion

Objectives


1. Explain 1-dimensional motion using the terms, position, velocity, and acceleration

2. Given an analytical expression for {position, velocity, acceleration} find expressions for the other two.

3. Given velocity, calculate total distance

4. Answer interpretative questions, such as:

a. What is the object’s {position, velocity, acceleration} at a given time?

b. What is the object’s speed at a given time?

c. At what times does the object have a given {position, velocity, acceleration}?

d. When is the object stopped?

e. When does the object turn around?

f. When is {position, velocity} maximized or minimized?

44. GDC required

A particle is moving along a straight line so that t seconds after passing through a fixed point O on the line, its velocity v

(t) m s–1

is given by

.

(a) Find the values of t for which v(t) = 0, given that 0 t 6. (3)

(b) (i) Write down a mathematical expression for the total distance travelled by the particle in the

first six seconds after passing through O.

(ii) Find this distance. (4)

(Total 7 marks)

45. GDC required

The displacement s metres of a moving body B from a fixed point O at time t seconds is given by

s = 50t – 10t2 + 1000.

(a) Find the velocity of B in m s–1

.

(b) Find its maximum displacement from O. (Total 6 marks)

46. No GDC allowed

The acceleration, a(t) m s–2

, of a fast train during the first 80 seconds of motion is given by

a(t) = – t + 2

where t is the time in seconds. If the train starts from rest at t = 0, find the distance travelled by the train in the first

minute.

(Total 4 marks)

tttv

3sin)(

201

Key topic #16: Related rates

Objectives


1. Recognize situations in which a change in one quantity W.R.T. time causes a related quantity to change W.R.T. time

2. Write equations that express the relationship between two or more quantities

3. Use implicit differentiation to determine how a change in one quantity W.R.T. time causes a related quantity to change W.R.T. time.

47. No GDC allowed

Air is pumped into a spherical ball which expands at a rate of 8 cm3 per second (8 cm

3 s

–1). Find the exact rate of

increase of the radius of the ball when the radius is 2 cm.

(Total 6 marks)

48. No GDC allowed

An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an

observer at ground level. At a given instant the observer notes that the angle is radians and is increasing at

radians per second. Find the speed, in kilometres per hour, at which the airplane is moving towards the observer.

(Total 6 marks)

3

1

60

1

x

3 km

Airplane

Observer

Key topic #17: Area and volume

Objectives


4. Find the area of plane regions:

a. Between a curve and the -axis

b. Between a curve and the -axis

c. Between two curves

5. Find the volume of solids generated when a plane region is rotated about:

a. The -axis

b. The -axis

49. No GDC allowed

Calculate the area bounded by the graph of y = x sin (x2) and the x-axis, between x = 0 and the smallest positive x-

intercept.

(Total 3 marks)

50. GDC required

Find the area of the region enclosed by the graphs of y = sin x and y = x2 – 2x + 1.5, where 0 x .

(Total 3 marks)

51. No GDC allowed

The area of the enclosed region shown in the diagram is defined by

y x2 + 2, y ax + 2, where a > 0.

This region is rotated 360° about the x-axis to form a solid of revolution. Find, in terms of a, the volume of this solid of

revolution.

(Total 4 marks)

x

y

a0

2

Key topic #18: Separable differential equations

Objectives


1. Solve separable differential equations:

a. For general solutions

b. For particular solutions given an initial condition

2. Explain the conditions that lead to exponential growth in terms of differential equations

52. No GDC allowed

When air is released from an inflated balloon it is found that the rate of decrease of the volume of the balloon is

proportional to the volume of the balloon. This can be represented by the differential equation = – kv, where v is the

volume, t is the time and k is the constant of proportionality.

(a) If the initial volume of the balloon is v0, find an expression, in terms of k, for the volume of

the balloon at time t.

(b) Find an expression, in terms of k, for the time when the volume is

(Total 4 marks)

53. GDC required

A sample of radioactive material decays at a rate which is proportional to the amount of material present in the sample.

Find the half-life of the material if 50 grams decay to 48 grams in 10 years.

(Total 3 marks)

54. No GDC allowed

Solve the differential equation xy = 1 + y2, given that y = 0 when x = 2.

(Total 3 marks)

tv

dd

.v

20

x

y

d

d

ib math hl: cumulative review 01irecognizethelionbyhispaw.com/ib/2012_hl.pdf · ib math hl:...

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