Back
Problem 4.1.52d
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
d. Determine all extrema of f.
Problem 4.7.109e
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫[f(x) + g(x)] dx
Problem 16
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y=1-(x+1)^3
Problem 23
The Mean Value Theorem
a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .
[Technology Exercises] b.Find the solution to as many decimal places as you can.
Problem 24
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y = (x² - 49) / (x² + 5x - 14)
Problem 25
As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.)
Problem 27
Finding Functions from Derivatives
Suppose that f(−1) = 3 and that f'(x) = 0 for all x. Must f(x) = 3 for all x? Give reasons for your answer.
Problem 28
Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?
Problem 29a
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
a. f(0) = 0
Problem 29b
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
b. f(1) = 0
Problem 30
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
30. y = (x² - 4) / (x² - 2)
Problem 31a
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y′ = x
Problem 31b
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
b. y′ = x²
Problem 33
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
33. y = (x² - x + 1) / (x - 1)
Problem 33a
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y′ = −1 / x²
Problem 34a
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y' = 1 / 2√x
Problem 35c
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
c. y' = sin (2t) + cos (t/2)
Problem 36
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
36. y = (x³ + x - 2) / (x - x²)
Problem 37
Graphs and Graphing
Graph the curves in Exercises 33–42.
y = 𝓍³ (8―𝓍 )
Problem 38
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
g'(x) = 1 / x² + 2x, P(−1, 1)
Problem 39
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
r'(θ) = 8 − csc²θ, P(π/4, 0)
Problem 40
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
r'(t) = sec t tan t − 1, P(0, 0)
Problem 41
Finding Position from Velocity or Acceleration
Exercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.
v = 9.8t + 5, s(0) = 10
Problem 42
Graphs and Graphing
Graph the curves in Exercises 33–42.
______
y = 𝓍√4 ― 𝓍²
Problem 43
Finding Position from Velocity or Acceleration
Exercises 41–44 give the velocity v = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.
v = sin πt, s(0) = 0
Problem 44
Each of Exercises 43–48 gives the first derivative of a function y = ƒ(𝓍). (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.
y' = 𝓍² ― 𝓍―6
Problem 45
Finding Position from Velocity or Acceleration
Exercises 45–48 give the acceleration a=d²s/dt², initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.
a = 32, v(0) = 20, s(0) = 5
Problem 46
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
46. y = cos(x) + √3 * sin(x), 0 ≤ x ≤ 2π
Problem 47
Each of Exercises 43–48 gives the first derivative of a function y = ƒ(𝓍). (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.
y' = 𝓍⁴ ― 2𝓍²
Problem 48
The range R of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity at an angle with the horizontal, then in Chapter 13 we find that R-v_0^2/g(sin 2α) where g is the downward acceleration due to gravity. Find the angle α for which the range R is the largest possible.