Question 1

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

Accepted Answer

To find the instantaneous rate of change of a function.

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}$$ using secant lines with points approaching x$$ = \frac{1}{2}$$. What value does the slope approach?

Accepted Answer

$$1$$

Question 4

If $$y = 40t - 16t^2$ gives the height (in feet) of a ball t$ seconds after being thrown, what is the average velocity between t = 2$ and t = 2.5$ seconds?

Accepted Answer

$$-32 ft/s$$

Question 5

For the same function $$y = 40t - 16t^2$, estimate the instantaneous velocity at t = 2$ seconds.

Accepted Answer

$$-24 ft/s$$

Question 6

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

Accepted Answer

A tangent line has the same direction as the curve at the point of contact.

Question 7

The area problem in calculus is primarily concerned with:

Accepted Answer

Finding the area of a region bounded by a function.

Question 8

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

Accepted Answer

$$x = -2$$

Question 9

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 10

A ball is thrown upward and its height is modeled by $$y = 40t - 16t^2$. Which of the following best describes the process for estimating the instantaneous velocity at t = 2$?

Accepted Answer

Calculate the average velocity over intervals that get closer to $$t = 2$$ and observe the limiting value.

Question 11

Which of the following is a correct interpretation of the average velocity of an object over the interval $$[a, b]$$?

Accepted Answer

The slope of the secant line connecting $$(a, f(a))$$ and $$(b, f(b)).$$

Question 12

Suppose $$f(x) = x^2$. What is the equation of the tangent line to f$ at x = 1$?

Accepted Answer

$$y = 2x - 1$$

Question 13

Which of the following real-world scenarios best illustrates the concept of instantaneous velocity?

Accepted Answer

Reading the speedometer of a car at a specific moment.

Question 14

If the slope of the secant line for $$f(x) = x^2$ between x = 0.5$ and x = 0.75$ is 1.25$, what does this value represent?

Accepted Answer

The average rate of change between $$x = 0.5$$ and $$x = 0.75.$$

Question 15

Which of the following best describes the relationship between secant and tangent lines as the two points on the secant line get closer together?

Accepted Answer

The secant line becomes the tangent line in the limit.

Introduction to Calculus: Limits, Tangent Lines, and Velocity

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