PART A: Solve the following equations/inequalities. Give ...

CFHS

Honors Precalculus Calculus BC Review

PART A: Solve the following equations/inequalities. Give all solutions.

1. 2x3 + 3x2 - 8x = -3

2. 3x-1 + 4 = 8

3.

1

2

5

x + 1 - x - 4 = x2 - 3x - 4

1 4. log2 2 + log2(x - 1) = 2

x 5. -4 cot = 4 3

3

6. sin 2x - sin x = 0

7. ln x - ln(x + 1) = 4

8. x3 + x + 2 = 0

9. 2 log x - log(x + 1) = -1

11

10. -

>0

x x-3

x

x

11. cos = tan x cos

2

2

1 12. 2 log3(x - 1) - 3 log3 27 = 1

13.

x3 2

-8=0

3

2

14. x +

+

0

x+2 x-2

1

15. log 1 (x - 1) - 2 log 1 4 = 4

32

2

16. 1 23x = 75x 16

sin 2x

17. cos x cos 2x +

=0

2

18. cos 2x + sin x = 0

19. sin 3x = sin x

20. cos 2x + 5 cos x = 2

21. sin2 x - 2 cos x - 3 = 0

22. cos 2x + cos 4x = 0 23. sin2 x = cos2 x

2 24. sin2 x = 2 cos2 x - 1

2 25. cos x sin 2x - 2 sin 2x = 0

PART B: For each of the following functions, rewrite the function in a way that makes it easier to graph, if necessary and graph the function.

? List all critical information for the function (x and y intercepts, domain, range, removable discontinuities, non-removable discontinuities, domain, range, end behavior (in limit notation), etc.)

1. f (x) = x5 - 5x4 + 11x3 - 23x2 + 28x - 12

2. f (x) = 3 ln 2 + ln(x - 2)

12

1

3. f (x) =

-+

x + 1 x x(x + 1)

4. f (x) = sin2 x + 2 sin(2x) cos(2x) + cos2 x

5. f (x) = x-1 + 3x-2

6. f (x) = exe2eln 3 + ln 1 + 4 ln e

x1 2

+

x

7. f (x) =

x

sin2 x 8. f (x) =

1 - cos x

- 20(x

-

1)

1 3

9. f (x) =

45

x2 + x - 2 10. f (x) =

x-1

cos x 1 + sin x

11. f (x) =

+

1 + sin x cos x

1x 12. f (x) = -

99

13. f (x) = 5x-1 + 1

5x-

1 3

14. f (x) =

x2 3

+1

15. f (x) = -2 sin(2x - ) - 1

2

x 16. f (x) = 3 tan - 3

2

17. f (x) = 4 csc(2x + ) + 8

18. f (x) = - arcsin(2x) +

19. f (x) = cos2 x

20. f (x) = x cos x x

21. f (x) = 1 + 2x - 3 sin 3

22. f (x) = 4 sin2 x - 4 sin x + 1

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HPC Calc BC Review

PART C 1. For the function f (x) = x2:

(a) Find the average rate of change of f on the interval [4, 6]. Sketch a graph that shows this. (b) Find the average rate of change of f on the interval [4, 4 + h]. Use this to find the slope of

the line tangent to f at x = 4. Sketch a graph that shows this. (c) Find the average rate of change of f on the interval [a, a + h]. Use this to find the slope of

the line tangent to f at x = a. Sketch a graph that shows this.

2. For the function f (x) = x:

(a) Find the average rate of change of f on the interval [4, 9]. Sketch a graph that shows this. (b) Find the average rate of change of f on the interval [4, 4 + h]. Use this to find the slope of

the line tangent to f at x = 4. Sketch a graph that shows this. (c) Find the average rate of change of f on the interval [a, a + h]. Use this to find the slope of

the line tangent to f at x = a. Sketch a graph that shows this.

1 3. For the function f (x) = :

x

(a) Find the average rate of change of f on the interval [4, 6]. Sketch a graph that shows this.

(b) Find the average rate of change of f on the interval [4, 4 + h]. Use this to find the slope of the line tangent to f at x = 4. Sketch a graph that shows this.

(c) Find the average rate of change of f on the interval [a, a + h]. Use this to find the slope of the line tangent to f at x = a. Sketch a graph that shows this.

4. Find the partial fraction decomposition of each of the following functions; use this to sketch its graph.

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

x + 17 (b) f (x) =

2x2 + 5x - 3

3x2 - 4x + 3 (c) f (x) =

x3 - 3x2 2x2 + x + 3 (d) f (x) =

x2 - 1

x3 + 2 (e) f (x) =

x2 - x

5. For each of the following functions h(x), find two functions f (x) and g(x) such that h(x) = f (g(x)). Neither function may be equal to x.

1 (a) h(x) =

(1 - x)2 (b) h(x) = tan(4x + 2)

(c) h(x) = 28(7x - 2)3 1

(d) h(x) = cos2(2x)

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HPC Calc BC Review

6. Given the tables of values, answer the following questions:

x f (x) -2 7 01 12

1 2

2

x g(x) -2 7 01 12 25

(a) Find the value of log4(f (2)) (b) Find the value of f -1(1)

(c) Find the value of f (g(1))

(d) Use transformations to move the point g(-2) = 7 to the corresponding point on h(x), given that h(x) = -3g(2x - 2) + 1

7. Given that f is a linear function such that f (-2) = 3 and f (0) = 1 and g(x) = tan(2x), answer the following:

(a) Find the value of ln(f (0))

(b) Find the value of g-1(1)

(c) Find the value of f (g())

(d) Use transformations to move the x-intercept of f to the corresponding point on h(x), given

f that h(x) =

-x 4

+1

2

8. The graph of f is shown. Draw the graph of each function.

(a) y = f (-x)

(b) y = -f (x)

(c) y = -2f (x + 1) + 1

(d) y = 3f (-2x - 4) - 1

9. Evaluate each of the following.

4 (a) sin

3 7 (b) tan 4

7 (c) sec

6

(d) cos-1

1 -

2

(e) sin-1 - 3

2 (f) tan(csc-1 (1))

10. Derive all double angle identities for sine, cosine, and tangent.

11. Derive all power reducing identities for sine, cosine, and tangent.

12. Derive all half angle identities for sine, cosine, and tangent.

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HPC Calc BC Review

13. Simplify the following expressions, rewriting without any trigonometric expressions; assume x > 0.

(a) tan(arcsin x)

x-4

(b) sin(arccos 6

14. Simplify and graph the following. (a) f (x) = (1 - 2 sin2 x)2 + 4 sin2 x cos2 x

(b) f (x) = 1 - 4 sin2 x cos2 x

15. Verify the following identities.

(a) csc x - cos x cot x = sin x

(b) 2 sin cos3 + 2 sin3 cos = sin 2

(c) sin 3x = (sin x)(3 - 4 sin2 x)

cos

sin

(d)

+

= cos + sin

1 - tan 1 - cot

cos(-x)

(e)

= 1 + sin(x)

sec(-x) + tan(-x)

3 tan x - 1 (f) tan x + =

4 1 + tan x

(g) cos2

x

1 + sec x =

2

2 sec x

(h) cos 4x = 1 - 8 sin2 x cos2 x

16. Use the Binomial Theorem to expand each of the following:

(a) (3x - 5)4

(b) (2 + 3y)5

(c) (-a + 4b)3

17. Determine the convergence or divergence of each sequence. If the sequence converges, find its limit.

3n + 1 (a) an = n

n (b) an = n + 1

(c) an = (1.1)n (d) 1, 1.5, 2.25, 3.375, ...

18. Write each of the following in sigma notation.

x3 x5 x7 x9 (a) x - + - + - . . .

3! 5! 7! 9! (b) x + x4 + x7 + x10 + . . .

(x - 2)2 (x - 2)3 (x - 2)4

(c) (x - 2) -

+

-

+...

2!

3!

4!

(d) 2(1.25)2 + 2(1.5)2 + 2(1.75)2 + 2(2)2 + 2(2.25)2 + 2(2.5)2 + 2(2.75)2 + 2(3)2

22

42

62

100 2

(e) 2 1 + + 2 1 + + 2 1 + + ? ? ? + 2 1 +

x

x

x

x

(x + 5) (x + 5)2 (x + 5)3 (x + 5)4

(f)

-

+

-

+...

1?2

2?3

3?4

4?5

1

2

3

4

n

(g) 3 2 + + 3 2 + + 3 2 + + 3 2 + + ? ? ? + 3 2 +

n

n

n

n

n

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HPC Calc BC Review

19. Evaluate the following, if possible. If it is not possible, explain why.

80

(a) (5 - n)

n=1

1k

(b) 3

k=1 2

40

(c) (j3 + 5j)

j=1

40

(d) (3j2 - 2)

j=1

5 (e)

n=0 4

2n 3

2 (f)

n=0 3

5n 4

j (g)

j=1 2

n

(h) (3j2 - 2j - 1)

j=1

n

(i) (2j + 3)

j=1

54

(j) (-5j + 2)

j=6

(x - 3)2 (y + 2)2

20. Give a set of parametric equations for the following ellipse:

+

=1

49

64

21. Eliminate the parameter for each of the following pairs of parametric equations (assume the domain for t is all real numbers unless specified otherwise). Graph the new function, and list any domain restrictions.

(a)

x = t2 y = 4t + 1

(b)

x = 4t - 3 y = 6t - 2

(c)

x = 2 cos t y = sec t

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