Download CFHS Honors Precalculus → Calculus BC Review PART A: Solve

Honors Precalculus → Calculus BC Review
CFHS
PART A: Solve the following equations/inequalities. Give all solutions.
1. 2x3 + 3x2 − 8x = −3
2. 3x−1 + 4 = 8
1
2
5
3.
−
= 2
x+1 x−4
x − 3x − 4
1
+ log2 (x − 1) = 2
2
x
√
5. −4 cot
=4 3
3
4. log2
6. sin 2x − sin x = 0
7. ln x − ln(x + 1) = 4
1
1
−
>0
x x−3
x
x
11. cos
= tan x cos
2
2
10.
1
12. 2 log3 (x − 1) − log3 27 = 1
3
9. 2 log x − log(x + 1) = −1
2
3
+
≤0
x+2 x−2
15.
1
log 1 (x − 1) − 2 log 1 4 = 4
2
2
3
16.
1 3x
2 = 75x
16
8. x3 + x + 2 = 0
sin 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
3
13. x 2 − 8 = 0
14. x +
17. cos x cos 2x +
22. cos 2x + cos 4x = 0
x
23. sin2 x = cos2
2
x
24. sin2
= 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.)
r
1 x
1. f (x) = x5 − 5x4 + 11x3 − 23x2 + 28x − 12
12. f (x) =
−
9 9
2. f (x) = 3 ln 2 + ln(x − 2)
13. f (x) = 5x−1 + 1
2
1
1
− +
3. f (x) =
1
x + 1 x x(x + 1)
5x− 3
14. f (x) =
+1
2
4. f (x) = sin2 x + 2 sin(2x) cos(2x) + cos2 x
x3
5. f (x) = x−1 + 3x−2
15. f (x) = −2 sin(2x −
6. f (x) = ex e2 eln 3 + ln 1 + 4 ln e
1
x2 + x
7. f (x) = √
x
16. f (x) = 3 tan
x
2
π
)−1
2
−3
17. f (x) = 4 csc(2x + π) + 8
2
sin x
1 − cos x
√
1
− 20(x − 1) 3
√
9. f (x) =
45
8. f (x) =
10. f (x) =
x2 + x − 2
x−1
cos x
1 + sin x
11. f (x) =
+
1 + sin x
cos x
18. f (x) = − arcsin(2x) + π
19. f (x) = cos2 x
√
20. f (x) = x cos x
21. f (x) = 1 + 2x − 3 sin
πx 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.
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 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
2
x + 2x
x + 17
(b) f (x) = 2
2x + 5x − 3
(a) f (x) =
3x2 − 4x + 3
x3 − 3x2
2x2 + x + 3
(d) f (x) =
x2 − 1
(c) f (x) =
(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
(c) h(x) = 28(7x − 2)3
1
(d) h(x) =
2
cos (2x)
(b) h(x) = tan(4x + 2)
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HPC → Calc BC Review
6. Given the tables of values, answer the following questions:
x f (x)
x g(x)
-2 7
-2 7
0 1
0 1
1 2
1 2
1
2 5
2
2
(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(π))
(d) Use transformations
to move the x-intercept of f to the corresponding point on h(x), given
f −x
4
+1
that h(x) =
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.
4π
7π
(a) sin
(c) sec
3
6
7π
1
(b) tan
(d) cos−1 −
4
2
(e) sin−1
(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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3
−
2
HPC → Calc BC Review
13. Simplify the following expressions, rewriting without any trigonometric expressions; assume x > 0.
√
x−4
(a) tan(arcsin x)
(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.
cos(−x)
= 1 + sin(x)
sec(−x) + tan(−x)
tan x − 1
3π
=
(f) tan x +
4
1 + tan x
x 1 + sec x
(g) cos2
=
2
2 sec x
(h) cos 4x = 1 − 8 sin2 x cos2 x
(a) csc x − cos x cot x = sin x
(e)
(b) 2 sin θ cos3 θ + 2 sin3 θ cos θ = sin 2θ
(c) sin 3x = (sin x)(3 − 4 sin2 x)
(d)
cos θ
sin θ
+
= cos θ + sin θ
1 − tan θ 1 − cot θ
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
n
n
(b) an =
n+1
(c) an = (1.1)n
(a) an =
(d) 1, 1.5, 2.25, 3.375, ...
18. Write each of the following in sigma notation.
x3 x5 x7 x9
+
−
+
− ...
3!
5!
7!
9!
(b) x + x4 + x7 + x10 + . . .
(a) x −
(x − 2)2 (x − 2)3 (x − 2)4
+
−
+ ...
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
2
2
2
2
4
6
100
2
(e) 2 1 +
+2 1+
+2 1+
+ ··· + 2 1 +
x
x
x
x
(c) (x − 2) −
(x + 5) (x + 5)2 (x + 5)3 (x + 5)4
−
+
−
+ ...
1×2
2×3
3×4
4×5
r
r
r
r
r
1
2
3
4
n
(g) 3 2 + + 3 2 + + 3 2 + + 3 2 + + · · · + 3 2 +
n
n
n
n
n
(f)
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HPC → Calc BC Review
19. Evaluate the following, if possible. If it is not possible, explain why.
(a)
80
P
(5 − n)
n=1
k
∞
P
1
(b)
3
2
k=1
(c)
40
P
(j 3 + 5j)
j=1
40
P
(h)
(3j 2 − 2)
j=1
n
∞
P
2
5
(e)
4
3
n=0
n
∞
P 2
5
(f)
3
4
n=0
∞
j
P π
(g)
j=1 2
(d)
n
P
(3j 2 − 2j − 1)
j=1
(i)
n
P
(2j + 3)
j=1
(j)
54
P
(−5j + 2)
j=6
20. Give a set of parametric equations for the following ellipse:
(x − 3)2 (y + 2)2
+
=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
(c)
π
x = 2 cos t
0<t<
y = sec t
2
(b)
x = 4t − 3
y = 6t − 2
(d)
x = 3 cos t
y = 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 every
three minutes. The bottom of the Ferris wheel is 6 feet off the ground. A rider enters at the
bottom 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 a
56 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 the
ground, and how far will it travel?
(b) Suppose a gust of wind blowing with the hammer at 10 feet per second occurs at the moment
it 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 weight
is released 5 feet above the ground at an angle of 85◦ with respect to the horizontal and the sheaf
reaches 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). Write
a 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) where
this occurs.
(a) r = 2 cos θ
(c) r = 5 sin(3θ)
(e) r = 3 − 6 cos θ
(b) r = 4 cos(2θ)
(d) r = 5 + 5 sin 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
(c) r = 1 + sin θ and r = 1 − cos θ
√
(d) r2 = 4 cos θ and r = 2
(b) r = 3 and r = 6 cos(2θ)
29. Write a set of parametric equations for the curve r = 4 cos(5θ).
30. Given the following pairs of vectors, find:
~ (sketch and component form)
• ~u − 2v
• |~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
√ √ (c) ~u = − 3, 1 , ~v = 2 3, 2
(a) ~u = h12, −5i, ~v = h3, 4i
(b) ~u = h2, −1i, ~v = h−2, −4i
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