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Problem 3
Sketch the graph of a function continuous on the given interval that satisfies the following conditions.
ƒ is continuous on the interval [-4, 4] ; f'(x) = 0 for x = -2, 0, and 3; ƒ has an absolute minimum at x = 3; ƒ has a local minimum at x = -2 ; ƒ has a local maximum at x = 0; ƒ has an absolute maximum at x = -4.
Problem 3.1.48
Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.
When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).
Problem 3.8.13a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x⁴+y⁴ = 2;(1,−1)
Problem 4.R.2e
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
e. On what intervals (approximately) is f concave up?
Problem 4.R.5b
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.
Problem 4.R.6
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ - 6x² on [-1, 5]
Problem 4.R.8
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x⁴ - 50x² on [-1, 5]
Problem 4.R.10
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ ln x on (0, ∞)
Problem 4.R.12
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = sin 2x + 3 on [-π , π]
Problem 4.R.14
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]
Problem 4.R.16
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]
Problem 4.R.2a
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
a . Give the approximate coordinates of the local maxima and minima of ƒ
Problem 4.R.2b
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
b. Give the approximate coordinates of the absolute maximum and minimum values of ƒ (if they exist).
Problem 4.R.2c
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
c. Give the approximate coordinates of the inflection point(s) of f.
Problem 4.R.2d
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
d. Give the approximate coordinates of the zero(s) of f.
Problem 4.R.2f
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
f. On what intervals (approximately) is f concave down?
Problem 4.R.5a
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.
Problem 4.R.5c
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
c. Determine where f has local maxima and minima.
Problem 4.R.61
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_t→0 (1 - cos 6t) / 2t
Problem 4.R.63
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
Problem 4.R.65
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 2Θ cot 3Θ
Problem 4.R.67
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_y→0⁺ (ln¹⁰ y) / √y
Problem 4.R.69
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3
Problem 4.R.71
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→0 csc x sin⁻¹ x
Problem 4.R.83
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
x¹⸍² and x¹⸍³
Problem 4.R.84
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
ln x and log₁₀ x
Problem 4.R.87
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
eˣ and 3ˣ
Problem 4.R.89
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
2ˣ and 4ˣ⸍²
Problem 4.R.116a
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cos xⁿ) / x²ⁿ
Problem 4.R.116b
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cosⁿ x) / x²