Solutions should show all of your work, not just a single nal answer. 3 ...
[Pages:7]Math 1131 Week 4 Worksheet
Name: Discussion Section: Solutions should show all of your work, not just a single final answer.
3.1: Derivatives of Polynomials and Exponential Functions
1. Use differentiation rules from Section 3.1 (not other methods) to compute the derivative of the following functions.
(a) f (x) = 7x3 - 5x + 8
(b) f (x) = ex + xe
(c) f (x) = 3x + 3x
(d)
f (x)
=
4x
-
4ex
x2 + 4x + 3 (e) f (x) =
x
12 7 (f) f (x) = -
x5 5 x
2. Use differentiation rules to find the equation of the tangent line to y = x2 - x4 (see below) at the point (1, 0). y
x 1
3. Use differentiation rules to find the values of a and b that make the function
x2 f (x) =
ax3 + bx
if x 2, if x > 2
differentiable at x = 2.
4. Find all points (c, f (c)) on the graph of f (x) = x3 - 3x2 where the tangent line has slope 9.
1
5. T/F (with justification) If f (x) = 7 for all x, then f (x) = for all x.
27
3.2: The Product and Quotient Rules
6. Compute the derivative of each function below using the methods from Sections 3.1 and 3.2 (not other methods).
x
(a) f (x) =
(simplify numerator in final answer)
x+3
ex
(b) f (x) =
(simplify numerator in final answer)
1 + ex
(c) f (x) = xex
ex (d) f (x) = xn for constant n, in two ways: (i) quotient rule and (ii) product rule
11
(e) f (x) = +
(in final answer, use a common denominator and simplify nu-
x 1-x
merator)
7. In the function h(x) below, defined in terms of f (x) and g(x), determine h (2) in each case if f (2) = 3, g(2) = 4, f (2) = 1, and g (2) = -5. (a) h(x) = 2f (x) + 5g(x)
(b) h(x) = f (x)g(x)
f (x) (c) h(x) =
g(x)
g(x) (d) h(x) =
f (x) + 2
Answers to selected problems
1. (a) f (x) = 21x2 - 5
(b) f (x) = ex + exe-1 3
(c) f (x) = 3 + 2x
(d) f (x) = 1 - 4ex. 4x3/4
3 2
3
(e) f (x) = 2
x
+
x
-
2x3/2 .
60 7 (f) f (x) = - + .
x6 5x6/5
2. y = -2x + 2
3. a = 1/4, b = 1.
4. (-1, f (-1)) = (-1, -4) and (3, f (3)) = (3, 0).
5. False
6.
(a)
3 (x+3)2
(b)
ex (1+ex)2
(c)
(2x+1)ex 2x
(d)
(i):
xn ex -ex nxn-1 x2n
(ii): exx-n - nexx-n-1 (show why these are the same!)
(e)
2x-1 x2(1-x)2
7. (a) -23
(b) -11
(c)
19 16
(d)
-
29 25
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