BackDifferential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the volume V = x³ of a cube when the edge lengths change from x₀ to x₀ + dx
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the volume V = (4/3)πr³ of a sphere when the radius changes from r₀ to r₀ + dr
Checking the Mean Value Theorem
Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.
f(x) =√(x − 1), [1, 3]
Checking the Mean Value Theorem
Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.
g(x) = {x³, −2 ≤ x ≤ 0
x², 0 < x ≤ 2
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = x²ᐟ³, [−1, 8]
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = x⁴ᐟ⁵, [0, 1]
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = √(x(1 − x)), [0, 1]
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {sinx / x, −π ≤ x < 0
0, x = 0
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {2x − 3, 0 ≤ x ≤ 2
6x − x² − 7, 2 < x ≤ 3
Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
f(x) = x⁴, x₀ = 1, dx = 0.1
Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
f(x) = x⁻¹, x₀ = 0.5, dx = 0.1
Estimating height of a building A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%?
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
a. surface area?
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
b. volume?