BackApproximation 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?
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x⁴ + 3x + 1, [−2, −1]
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
g(t) = √t + √(1 + t) − 4, (0, ∞)
"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
r(θ) = 2θ − cos²θ + √2, (−∞, ∞)
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.
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
a. f(0) = 0
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
b. f(1) = 0