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Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Week in Review 5
Sections 3.2 & 3.4
1. Differentiate the following functions.
√
1
(a) f (x) = 9x + 3 x + √
+ π2
53x
(b) f (x) = (5x5 − 7x3 + 9x +
(c) g(t) =
4t4 + 3t − 2
√
t
(d) h(x) =
5x2 + 7x
√
x3 − 4 9
√
5)(3x6 − 10x2 + e5 + cos 3)
1
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
2. Given that f (2) = 5 and f 0 (2) = 1, find g 0 (2) if g(x) = (x3 + 1)(f (x) + 5x).
3. Consider the function f (x) = 2x(x2 + 1).
(a) Find the values of x for which the tangent line to the graph of f is parallel to the line 8x − 2y = 9.
(b) For what values of a and b is the line ax + by = 6 tangent to the graph of f at x = 1?
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
4. Find the equation of the tangent line to the graph of f (x) =
x2
x
at x = 1.
+5
5. At what points on the graph of f (x) = −x2 + 4 does the tangent line also pass through the point
(1, 7)?
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
6. Find f 0 (x) for the function below. Where is f not differentiable?
f (x) =


 4x + 11
6−
x2

 −2x + 6
if x ≤ −2
if − 2 < x < 2
if x ≥ 2
7. Given f (x) below, find the values of a and b that make f differentiable everywhere.
(
f (x) =
ax + b
x2 − x
if x ≤ 3
if x > 3
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
8. Calculate the following limits.
(a) lim
sin 9x
x(cos x + 1)
(b) lim
(cos x − 1) sin 5x
x2
x→0
x→0
cot 3x
x→0 csc 4x
(c) lim
tan2 3x
x→0 6x2
(d) lim
5
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
9. Find the derivatives of the following functions.
(a) f (x) = sec x cot x + csc x
(b) g(x) =
x − cos x
tan x + sin x
10. Find the tangent line to the graph of f (x) = tan x + 4 at x = π4 .
11. For what values of x does f (x) = sin x − cos x have a horizontal tangent line?
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