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Math 11-Worksheet 05 Sol
Course: Calculus I (MATH 011)
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University: University of California, Merced
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Math 11 Worksheet 5 for Section 3.3 (Solutions) Spring 2023
1 Section 3.3
1. Given the following information, compute the indicated derivatives.
f′(x) = 2
3x2+ 1 dg
dx = 2 −xdh
dx =−5p′(x) = 1
x
(a) d
dx [f(x)−g(x)] =2
3x2+x−1
(b) 2
5h(x) + 3p(x)′
=−2 + 3
x
(c) d
dx 3
2f(x) + 2g(x) + 5h(x)−p(x) + 6
=x2−2x−39
2−1
x
2. Compute the derivatives of the following functions.
(a) A(x) = x11
A′(x) = 11x10
(b) B(x) = 3x2−4x+ 1
B′(x) = 6x−4
(c) C(p) = 2 −4p2+ 6p5
C′(x) = −8p+ 30p4
(d) D(t) = −t3+ 4t2−t
t
D′(x) = −2t+ 4
(e) E(x) = ax +b
c
(where a, b, c are constants)
E′(x) = a
c
3. Given the following functions, find the equation of the tangent line at the given point.
(a) f(x) = 2x2−3x+ 1, x = 1
Slope: f′(1) = 1, Passing through: (1,0), Tangent: y=x−1
(b) g(x) = 1 + 4x2−x3, x =−1
Slope: g′(−1) = −11, Passing through: (−1,6), Tangent: y=−11x−5
(c) h(x) = −3 + x2−2x4, x = 0
Slope: h′(0) = 0, Passing through: (0,−3), Tangent: y=−3
4. Determine all xvalues that the given function is horizontal.
(a) f(x) = x3+ 3x2−5
f′(x) = 3x2+ 6x. Setting f′(x) = 0 yields x= 0,−2.
(b) g(x) = −8x3+ 10x2+ 4x−42
g′(x) = −24x2+ 20x+ 4 = 0. Setting g′(x) = 0 yields x= 1,−1/6.
(c) h(x) = x4−8x2
h′(x) = 4x3−16x. Setting h′(x) = 0 yields x= 0,±2.
5. Find all xvalues that the given function has the indicated slope.
(a) f(x) = −3x2+ 2x, m = 1
f′(x) = −6x+ 2. Setting f′(x) = 1 yields x= 1/6.
(b) g(x) = x3+ 3x2−5, m =−3
g′(x) = 3x2+ 6x. Setting g′(x) = −3 yields x=−1.
(c) h(x) = 1
3x3+x2−x, m = 2
h′(x) = x2+ 2x−1. Setting h′(x) = 2 yields x=−3,1.
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