BackThe accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after it has been driven for t days.
c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.
Existence of Limits
In Exercises 5 and 6, explain why the limits do not exist.
limx→0 x/|x|
Suppose that a function f(x) is defined for all real values of x except x=c. Can anything be said about the existence of limx→c f(x)? Give reasons for your answer.
Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.
If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = x², x = -2
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = 3x - 4, x = 2
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = 1/x, x = -2
Using the Sandwich Theorem
If √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).
Using the Sandwich Theorem
If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
f(x)=x³+1
a. [2, 3]
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(x)=x²−2x
a. [1, 3]
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
h(t)=cot t
a. [π/4,3π/4]
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(t)=2+cos t
b. [0,π]
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
P(θ)=θ³ − 4θ² + 5θ; [1,2]