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문제 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.
문제 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
문제 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
문제 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.
문제 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)
문제 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.)
문제 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.
문제 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?
문제 29a
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
a. f(0) = 0
문제 29b
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
b. f(1) = 0
문제 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)
문제 31a
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y′ = x
문제 31b
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
b. y′ = x²
문제 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)
문제 33a
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y′ = −1 / x²
문제 34a
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y' = 1 / 2√x
문제 35c
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
c. y' = sin (2t) + cos (t/2)
문제 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)
문제 37
Graphs and Graphing
Graph the curves in Exercises 33–42.
y = 𝓍³ (8―𝓍 )
문제 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)
문제 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)
문제 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)
문제 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
문제 42
Graphs and Graphing
Graph the curves in Exercises 33–42.
______
y = 𝓍√4 ― 𝓍²
문제 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
문제 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
문제 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
문제 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
문제 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
문제 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.