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

x+ 1− 2

x− 4=

5

x2 − 3x− 4

4. log2

1

2+ log2(x− 1) = 2

5. −4 cot(x

3

)= 4√

3

6. sin 2x− sinx = 0

7. lnx− ln(x+ 1) = 4

8. x3 + x+ 2 = 0

9. 2 log x− log(x+ 1) = −1

10.1

x− 1

x− 3> 0

11. cos(x

2

)= tanx cos

(x2

)12. 2 log3(x−1)− 1

3log3 27 = 1

13. x32 − 8 = 0

14. x+3

x+ 2+

2

x− 2≤ 0

15.1

3log 1

2(x− 1)− 2 log 1

24 = 4

16.1

1623x = 75x

17. cosx cos 2x+sin 2x

2= 0

18. cos 2x+ sinx = 0

19. sin 3x = sinx

20. cos 2x+ 5 cosx = 2

21. sin2 x− 2 cosx− 3 = 0

22. cos 2x+ cos 4x = 0

23. sin2 x = cos2(x

2

)24. sin2

(x2

)= 2 cos2 x− 1

25. cosx sin 2x− 2 sin 2x = 0

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

• List all critical information for the function (x and y intercepts, domain, range, removable dis-continuities, 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)

3. f(x) =1

x+ 1− 2

x+

1

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

7. f(x) =x

12 + x√x

8. f(x) =sin2 x

1− cosx

9. f(x) =−√

20(x− 1)13

√45

10. f(x) =x2 + x− 2

x− 1

11. f(x) =cosx

1 + sin x+

1 + sin x

cosx

12. f(x) =

√1

9− x

9

13. f(x) = 5x−1 + 1

14. f(x) =5x−

13

x23

+ 1

15. f(x) = −2 sin(2x− π

2)− 1

16. f(x) = 3 tan(x

2

)− 3

17. f(x) = 4 csc(2x+ π) + 8

18. f(x) = − arcsin(2x) + π

19. f(x) = cos2 x

20. f(x) =√x cosx

21. f(x) = 1 + 2x− 3 sin(πx

3

)22. f(x) = 4 sin2 x− 4 sinx+ 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 ofthe 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 ofthe 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 ofthe 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 ofthe line tangent to f at x = a. Sketch a graph that shows this.

3. For the function f(x) =1

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 ofthe 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 ofthe 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 itsgraph.

(a) f(x) =1

x2 + 2x

(b) f(x) =x+ 17

2x2 + 5x− 3

(c) f(x) =3x2 − 4x+ 3

x3 − 3x2

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

x2 − 1

(e) f(x) =x3 + 2

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.

(a) h(x) =1

(1− x)2

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

(c) h(x) = 28(7x− 2)3

(d) h(x) =1

cos2(2x)

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HPC → Calc BC Review

6. Given the tables of values, answer the following questions:x f(x)-2 70 11 2

21

2

x g(x)-2 70 11 22 5

(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), giventhat 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), answerthe following:

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

(b) Find the value of g−1(1)

(c) Find the value of f(g(π))

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

that h(x) =f(−x

4

)2

+ 1

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.

(a) sin

(4π

3

)(b) tan

(7π

4

) (c) sec

(7π

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)(b) sin(arccos

(√x− 4

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) cscx− cosx cotx = sinx

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

(c) sin 3x = (sinx)(3− 4 sin2 x)

(d)cos θ

1− tan θ+

sin θ

1− cot θ= cos θ + sin θ

(e)cos(−x)

sec(−x) + tan(−x)= 1 + sin(x)

(f) tan

(x+

3π

4

)=

tanx− 1

1 + tan x

(g) cos2(x

2

)=

1 + sec x

2 secx

(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 itslimit.

(a) an =3n+ 1

n

(b) an =n

n+ 1

(c) an = (1.1)n

(d) 1, 1.5, 2.25, 3.375, ...

18. Write each of the following in sigma notation.

(a) x− x3

3!+x5

5!− x7

7!+x9

9!− . . .

(b) x+ x4 + x7 + x10 + . . .

(c) (x− 2)− (x− 2)2

2!+

(x− 2)3

3!− (x− 2)4

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

(e) 2

(1 +

2

x

)2

+ 2

(1 +

4

x

)2

+ 2

(1 +

6

x

)2

+ · · ·+ 2

(1 +

100

x

)2

(f)(x+ 5)

1× 2− (x+ 5)2

2× 3+

(x+ 5)3

3× 4− (x+ 5)4

4× 5+ . . .

(g) 3

√2 +

1

n+ 3

√2 +

2

n+ 3

√2 +

3

n+ 3

√2 +

4

n+ · · ·+ 3

√2 +

n

n

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HPC → Calc BC Review

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

(a)80∑n=1

(5− n)

(b)∞∑k=1

3

(1

2

)k

(c)40∑j=1

(j3 + 5j)

(d)40∑j=1

(3j2 − 2)

(e)∞∑n=0

(5

4

)(2

3

)n

(f)∞∑n=0

(2

3

)(5

4

)n

(g)∞∑j=1

(π2

)j

(h)n∑

j=1

(3j2 − 2j − 1)

(i)n∑

j=1

(2j + 3)

(j)54∑j=6

(−5j + 2)

20. Give a set of parametric equations for the following ellipse:(x− 3)2

49+

(y + 2)2

64= 1

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

(a)x = t2

y = 4t+ 1

(b)x = 4t− 3y = 6t− 2

(c)x = 2 cos ty = sec t

0 < t <π

2

(d)x = 3 cos ty = 2 sin t+ 1

22. Give two different sets of parametric equations that for a line segment starting at the point(2, -3) and ending at the point (18, 9).

23. A Ferris wheel with a diameter of 30 feet rotates counterclockwise and makes one revolution everythree minutes. The bottom of the Ferris wheel is 6 feet off the ground. A rider enters at thebottom of the wheel.

(a) Write a set of parametric equations modeling the position of the rider after t minutes.

(b) Describe the position of the rider after five minutes.

24. In an event in the Highland Games Competition, a participant in the ‘hammer throw’ throws a56 pound weight for distance. The weight is released 6 feet above the ground at an angle of 42◦

with respect to the horizontal with an initial speed of 33 feet per second.

(a) Find the parametric equations for the flight of the hammer. When will the hammer hit theground, and how far will it travel?

(b) Suppose a gust of wind blowing with the hammer at 10 feet per second occurs at the momentit was tossed. How far does it travel now?

25. In another event, the ‘sheaf toss’, a participant throws a 20 pound weight for height. If the weightis released 5 feet above the ground at an angle of 85◦ with respect to the horizontal and the sheafreaches a maximum height of 31.5 feet, find how fast the sheaf was launched in the air.

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HPC → Calc BC Review

26. A particle moves along a horizontal line so that its position at any time t is given by s(t). Writea description of the motion.

(a) s(t) = −t2 + 3t, − 2 ≤ x ≤ 4 (b) s(t) = −t3 − 5t2 + 4t, − 5 ≤ x ≤ 2

27. Sketch the following polar equations. If the graph crosses the pole, indicate the angle(s) wherethis occurs.

(a) r = 2 cos θ

(b) r = 4 cos(2θ)

(c) r = 5 sin(3θ)

(d) r = 5 + 5 sin θ

(e) r = 3− 6 cos θ

(f) r = 2 + 4 sin(3θ)

28. Find the points of intersection of the following polar equations.

(a) r = 2 sin θ and r = 2− 2 sin θ

(b) r = 3 and r = 6 cos(2θ)

(c) r = 1 + sin θ and r = 1− cos θ

(d) r2 = 4 cos θ and r =√

2

29. Write a set of parametric equations for the curve r = 4 cos(5θ).

30. Given the following pairs of vectors, find:

• ~u− ~2v (sketch and component form)

• |~u| and |~v|• A vector in the direction of ~u with the same magnitude as ~v

• The dot product of ~u and ~v ; what does this mean about the angle between the vectors?

• The angle between the vectors

(a) ~u = 〈12,−5〉, ~v = 〈3, 4〉(b) ~u = 〈2,−1〉, ~v = 〈−2,−4〉

(c) ~u =⟨−√

3, 1⟩, ~v =

⟨2√

3, 2⟩

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