Q: The instantaneous velocity at $$t = 1$$ for $$s(t) = -16t^2$ + 96t is $$given by which limit?

$$\lim\limits_{t \to 1} \frac{s(t) - s(1)}{t - 1}$$

Question 1

Given the position function $$s(t) = -16t^2$ + 96t$$, what is the average velocity of the rock between $$t = 2$$ and $$t = 4$$ seconds?

Accepted Answer

$$0 ft/s$$

Question 2

Which of the following best describes the difference between average velocity and instantaneous velocity?

Accepted Answer

Average velocity is calculated over an interval, while instantaneous velocity is the limit of average velocities as the interval shrinks to a point.

Question 3

For the function $$f(x)$$, the difference quotient $$\frac{f(x) - f(a)}{x - a}$$ represents:

Accepted Answer

The average rate of change between $$x = a$$ and $$x$$

Question 4

If $$s(t) = -16t^2$ + 96t$$, estimate the instantaneous velocity at $$t = 2$$ using the average velocity over $$[2, 2.01].$$

Accepted Answer

$$31.68 ft/s$$

Question 5

Which statement is true about the tangent line to the curve $$y = f(x) at$$ $$x = c$$?

Accepted Answer

It passes through $$(c, f(c))$$ and has the same slope as the curve at that point.

Question 6

The instantaneous velocity at $$t = 1$$ for $$s(t) = -16t^2$ + 96t is $$given by which limit?

Accepted Answer

$$\lim\limits_{t 	o 1} \frac{s(t) - s(1)}{t - 1}$$

Question 7

A car's position is modeled by $$s(t) = -16t^2$ + 96t. $$What is the physical meaning of the slope of the secant line between $$t = 1$$ and $$t = 3$$?

Accepted Answer

The car's average velocity over $$[1, 3]$$

Question 8

If the average velocity over $$[1, 1.0001] is$$ $$63.9984 ft/s$$, what does this suggest about the instantaneous velocity at $$t = 1$$?

Accepted Answer

It is approximately $$64 ft/s$$

Question 9

Which of the following best expresses the connection between limits and the slope of the tangent line?

Accepted Answer

The slope of the tangent line is the limit of the slopes of secant lines as the interval shrinks to a point.

Question 10

Suppose $$f(x) is $$continuous and differentiable. Which of the following is the best estimate for the slope of the tangent line at $$x = a$$?

Accepted Answer

$$\frac{f(a + h) - f(a)}{h}$$ for very small $$h$$

Question 11

A ball is thrown upward and its position is given by $$s(t) = -16t^2$ + 96t. $$What is the instantaneous velocity at $$t = 0$$?

Accepted Answer

$$96 ft/s$$

Question 12

Why does the average velocity over smaller and smaller intervals around $$t = 1$$ approach the instantaneous velocity at $$t = 1$$?

Accepted Answer

Because the secant line approaches the tangent line at $$t = 1.$$

Question 13

If $$s(t) = -16t^2$ + 96t$$, what is the slope of the tangent line to the graph at $$t = 1$$?

Accepted Answer

$$64 ft/s$$

Question 14

Which of the following is NOT a correct interpretation of the limit $$\lim\limits_{t 	o 1} \frac{s(t) - s(1)}{t - 1}$$?

Accepted Answer

The area under the curve from $$t = 0 to$$ $$t = 1$$

Question 15

A rock is launched vertically. If the average velocity over $$[1, 1.1] is$$ $$62.4 ft/s$$, what does this value represent?

Accepted Answer

The average rate of change of position between $$t = 1$$ and $$t = 1.1$$

Chapter 2: Limits – The Foundation of Calculus

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