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Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Week in Review 11
Sections 5.1, 5.2, & 5.3
1. Given the graph of f ′ below, find the following. Then sketch a graph of f if f is continuous and
f (−6) = 0.
(i) The intervals where f is increasing and decreasing.
(ii) The x-values of any local extrema of f .
(iii) The intervals where f is concave up and concave down.
(iv) The x-values of any inflection points of f .
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
2. Sketch a graph of a function f with the following properties:
(i) lim− f (x) = ∞, lim+ f (x) = −∞
x→3
x→3
(ii) lim f (x) = 0
x→±∞
(iii) f is continuous everywhere except at x = 3.
(iv) f (0) = −3, f (7) = 3
(v) f ′ (x) > 0 on (−2, 3) ∪ (3, 7)
(vi) f ′ (x) < 0 on (−∞, −2) ∪ (7, ∞)
(vii) f ′′ (x) > 0 on (−5, 3) ∪ (5, 7) ∪ (7, ∞)
(viii) f ′′ (x) < 0 on (−∞, −5) ∪ (3, 5)
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
3. Find all critical values for the following functions.
(a) f (x) = x3 − 7x2 + 4
1
(b) f (x) = x 3 (x − 1)2
(c) f (x) =
ex
x+5
3
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(d) f (x) = |x2 − 5x|
4. Find all absolute and local extrema for the following functions by graphing.
(a) f (x) = x2 − 3, −1 ≤ x < 2
(b) f (x) =
(
x+2
−(x − 1)2
if − 2 < x ≤ 0
if 0 < x < 3
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
5. Find the absolute maximum and minimum values for the following functions on the given interval.
(a) f (x) = x4 − 8x2 + 1 on [−1, 3]
(b) f (x) = x + 2 sin x on [0, π]
(c) f (x) =
√
3
x2 − 6x on [−1, 4]
5
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(d) f (x) =
1
x−2
on [1, 3]; on (3, 4]
6. Sketch a general graph with the following properties.
(a) x = 3 is a critical number of f , but f has no local extrema.
(b) f is continuous and has a local minimum at x = 3, but f ′ (3) does not exist.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
7. Find the value of c that satisfies the Mean Value Theorem for the function f (x) = 3 − x2 on the
interval [1, 6].
8. Given that f (2) = 5 and that −4 ≤ f ′ (x) ≤ 4 for all x in the interval [−1, 2], what are the largest and
smallest values of f (−1)?
9. Find the intervals where the following functions are increasing/decreasing and identify all local extrema.
(a) f (x) = xe−2x
2
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(b) f (x) =
x+3
(x − 2)2
(c) f (x) = (x − π2 ) cos x − sin x on [0, 2π]
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
10. Find the intervals where the following functions are concave up and concave down and identify all
inflection points.
(a) f (x) = ex (x3 − 8x2 + 27x − 38)
(b) f (x) = x ln(x − 3)
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
11. Determine all intervals where the function f (x) = x6 − 6x5 is increasing and decreasing, concave up
and concave down. Identify all local extrema and points of inflection. Then, sketch a graph.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
12. Given the following information, identify any local extrema.
x
−3
4
6
8
f (x)
1
5
3
4
f ′ (x)
0
0
0
DNE
f ′′ (x)
−4
0
12
DNE
11