Question 1

Which of the following best describes the tangent line problem in calculus?

Accepted Answer

Finding the instantaneous rate of change of a function at a point.

Question 2

Given the function $$f(x) = x^2$, what is the slope of the secant line between the points $$\left(\frac{1}{2}, \frac{1}{4}\right)$$ and (1$$, 1)$$?

Accepted Answer

$$1.5$$

Question 3

For the function $$f(x) = x^2$, estimate the slope of the tangent line at x$$ = \frac{1}{2}$$ using the secant line between x = 0.5$ and x = 0.75$.

Accepted Answer

$$1.25$$

Question 4

If $$f(t)$$ describes the position of an object at time $$t$$, what does the slope of the tangent line to the graph of $$f at$$ $$t = a$$ represent?

Accepted Answer

The instantaneous velocity at time $$t = a.$$

Question 5

A ball is thrown into the air with a velocity of $$40 ft/s. $$Its height after $$t$$ seconds is given by $$y = 40t - 16t^2$. What is the average velocity between t = 2$ and t = 2.5$ seconds?

Accepted Answer

$$-32 ft/s$$

Question 6

For the function $$y = 40t - 16t^2$, estimate the instantaneous velocity at t = 2$ seconds using the average velocity over the interval $$[2, 2.01]$$.

Accepted Answer

$$-24.16 ft/s$$

Question 7

Which of the following statements about secant and tangent lines is correct?

Accepted Answer

A tangent line touches a curve at exactly one point and has the same direction as the curve at that point.

Question 8

Given $$f(x) = x^2$ + 4x - 1$$, at what $$x-$$value does the function have a tangent line with a slope of zero?

Accepted Answer

$$x = -2$$

Question 9

If the average velocity of a falling object over the interval $$[2, 2.1] is$$ $$-25.6 ft/s$$, what does this value represent?

Accepted Answer

The object's average velocity between $$t = 2$$ and $$t = 2.1.$$

Question 10

Which of the following best describes the area problem in calculus?

Accepted Answer

Finding the area of a region bounded by a function.

Question 11

Suppose $$f(x) = x^2$. What is the estimated slope of the tangent line at x = 0.5$ using the secant line between x = 0.5$ and x = 0.625$?

Accepted Answer

$$1.125$$

Question 12

A car's position is given by $$s(t) = t^2$ + 4t - 1. At $$what time does the car momentarily stop (i.e., when is its instantaneous velocity zero)?

Accepted Answer

$$t = -2$$

Question 13

If the slope of the secant line for $$f(x) = x^2$ between x = 0.5$ and x = 0.53125$ is 1.03125$, what does this value approximate?

Accepted Answer

The slope of the tangent line at $$x = 0.5$$

Question 14

A biologist models the population of a bacteria colony by $$P(t) = t^2$ + 4t - 1$$, where $$t is in $$hours. At what time does the population growth rate become zero?

Accepted Answer

$$t = -2$$

Question 15

Which of the following is a real-world example of the tangent line problem?

Accepted Answer

Calculating the speed of a car at a specific instant using its position function.

Introduction to Calculus: Limits, Tangent Lines, and Velocity

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