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Problem 3.2.45a
Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>
a. At which points is the slope of the curve negative?
Problem 3.2.19a
Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (-2,2) at which f is not continuous.
Problem 3.2.20b
Use the graph of g in the figure to do the following. <IMAGE>
b. Find the values of x in (-2,2) at which g is not differentiable.
Problem 3.2.73a
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
a. f(x) = (x-2)^1/3
Problem 3.2.73c
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
c. f(x) = √|x-4|
Problem 3.2.76
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. x²+y² = 9
Problem 3.2.48
Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>
Problem 3.23
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(x) = 5x³
Problem 3.25
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
h(t) = t²/2 + 1
Problem 3.26
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
Problem 3.28
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
g(t) = 6√t
Problem 3.3.56
Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
y = (x2 - 2ax + a2) / (x - a); a is a constant.
Problem 3.3.58
Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
r(t) = (e2t + 3et + 2) / (et + 2)
Problem 3.3.7
Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).
Problem 3.3.70
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x2 + 5ex
Problem 3.3.74b
Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x).
Find an equation of the line tangent to y = h(x) at x = 3.
Problem 3.3.89
Calculator limits Use a calculator to approximate the following limits.
lim x🠂0 e^3x-1 / x
Problem 3.30
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(s) = √s/4
Problem 3.32
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
g(x) = 6x⁵ - 5/2 x² + x + 5
Problem 3.33a
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3
Problem 3.36
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
s(t) = 4√t - 1/4t⁴+t+1
Problem 3.37a
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/ √x; a= 1/4
Problem 3.38
9–61. Evaluate and simplify y'.
y = (v / v+1)^4/3
Problem 3.39a
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √2x+1; a= 4
Problem 3.4.46
Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)
Problem 3.4.89
Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>
d/dx (xg(x)) | x=2
Problem 3.4.91a
The line tangent to the curve y=h(x) at x=4 is y = −3x+14. Find an equation of the line tangent to the following curves at x=4.
y = (x²-3x)h(x)
Problem 3.4.18
Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).
Problem 3.4.19
Derivatives Find and simplify the derivative of the following functions.
f(x) = 3x⁴(2x²−1)
Problem 3.4.23
Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t