Applications of Derivatives Online Quiz Questions

A circular ripple has radius increasing at 2 cm/s. At the instant when its radius is 5 cm, how fast is its area increasing?

Select all statements that are consistent with the definitions of velocity and speed for motion along a line.

True or false: If a function is continuous on [a, b] and differentiable on (a, b), then at least one point c in (a, b) has derivative equal to the function's average rate of change on [a, b].

What is the calculus term for the tangent-line approximation L(x) = f(a) + f'(a)(x − a) near x = a?

An object has position s(t) = t³ − 6t² + 9t for 0 ≤ t ≤ 4. At which times does it change direction?

For f(x) = x² on [1, 3], enter the value of c guaranteed by the Mean Value Theorem. Enter a single number; no tolerance is allowed.

Select all steps that belong to the closed-interval method for finding absolute extrema of a continuous function on [a, b].

True or false: If f' changes from negative to positive at a critical number c, then f has a local minimum at c.

Explain a reliable general procedure for solving a related-rates problem. Then apply it to a circular ripple whose radius increases at 2 cm/s when its radius is 5 cm, finding and interpreting the rate at which its area changes.

Using the linearization of f(x) = √x at a = 4, which value estimates √4.1?

Which procedure correctly avoids treating a changing quantity as a constant in a related-rates problem?

True or false: If a function is continuous on a closed interval [a,b], then it must attain both an absolute maximum and an absolute minimum on that interval.

An object's velocity at an instant is -7 m/s. What is its speed at that instant?

What is the mathematical name for the tangent-line approximation L(x)=f(a)+f'(a)(x-a) of a differentiable function near x=a?

Using linearization at x=4, what is the tangent-line estimate for √4.1?

A circular ripple has radius 5 cm and its radius is increasing at 2 cm/s. At what numerical rate is its area increasing? Enter the value in cm²/s, rounded to two decimal places; an absolute tolerance of 0.01 cm²/s is allowed.

For f(x)=x² on [1,3], which value of c satisfies the Mean Value Theorem equation f'(c)=[f(3)-f(1)]/(3-1)?

Suppose f'(c)=0 and f''(c)=0. What conclusion does the second derivative test provide about whether f has a local maximum or minimum at c?