Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


i. y = x² − 4

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.

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, ∞)