Q: Instantaneous velocity at time $$t_0 is $$defined as:

$$\lim_{\Delta t \to 0} \frac{s(t_0$ + \Delta t) - s(t_0$)}{\Delta t}$$

Q: A rock is launched vertically upward from the ground with a speed of 96 ft/s. Its position after $$t$$ seconds is $$s(t) = 96t - 16t^2$. What is the instantaneous velocity at t = 2$?

$$s'(2) = 96 - 16 \times 2 \times 2 = 32 ft/s$$

Question 1

A city bus passes milepost 100 miles at noon and milepost 125 miles at 12:30 PM. What is the average velocity of the bus during this interval?

Accepted Answer

$$\frac{125 - 100}{12.5 - 12} = 25 mi/hr$$

Question 2

Which device is commonly used in vehicles to measure instantaneous speed?

Accepted Answer

Speedometer

Question 3

Given $$s(t) = 16t^2$ represents the position of a rock (in feet) at time t$ (in seconds), what is the average velocity between t = 1$ and t = 3$?

Accepted Answer

$$\frac{16(3)^2$ - 16(1)^2$}{3 - 1} = \frac{144 - 16}{2} = 32 ft/s$$

Question 4

Instantaneous velocity at time $$t_0 is $$defined as:

Accepted Answer

$$\lim_{\Delta t 	o 0} \frac{s(t_0$ + \Delta t) - s(t_0$)}{\Delta t}$$

Question 5

The slope of the tangent line to the position-time graph at a point represents:

Accepted Answer

Instantaneous velocity at that point

Question 6

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$a. If$$ $$f(x) is $$arbitrarily close to $$L$$ for all $$x$$ sufficiently close (but not equal) to $$a$$, we write:

Accepted Answer

$$\lim_{x 	o a} f(x) = L$$

Question 7

Given the function $$f(x) = \frac{\sqrt{x} - 1}{x - 1}$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$0.5$$

Question 8

If the left-sided limit and right-sided limit of $$f(x) as$$ $$x$$ approaches $$a$$ are not equal, what can be concluded about $$\lim_{x 	o a} f(x)$$?

Accepted Answer

The two-sided limit does not exist

Question 9

A rock is launched vertically upward from the ground with a speed of 96 ft/s. Its position after $$t$$ seconds is $$s(t) = 96t - 16t^2$. What is the instantaneous velocity at t = 2$?

Accepted Answer

$$s'(2) = 96 - 16 	imes 2 	imes 2 = 32 ft/s$$

Question 10

Which of the following best describes the relationship between the average velocity and the slope of the secant line?

Accepted Answer

Average velocity equals the slope of the secant line between two points

Question 11

Given the graph of $$f(x)$$, if $$\lim_{x 	o 2^-} f(x) = 3$$ and $$\lim_{x 	o 2^+} f(x) = 5$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

Does not exist

Question 12

Suppose $$h(x) is $$defined as shown in the graph, and $$h(2) = 5$$, $$\lim_{x 	o 2} h(x) = 3. $$What does this indicate about $$h(x) at$$ $$x = 2$$?

Accepted Answer

There is a removable discontinuity at $$x = 2$$

Question 13

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$

Question 14

A car travels along a straight road. Its position at time $$t is $$given by $$s(t) = 4t^2$ + 2t. $$What is the average velocity between $$t = 1$$ and $$t = 3$$?

Accepted Answer

$$\frac{4(3)^2$ + 2(3) - [4(1)^2$ + 2(1)]}{3 - 1} = \frac{36 + 6 - (4 + 2)}{2} = \frac{42 - 6}{2} = 18$$ units/time

Question 15

Given $$f(x) = x^2$ and a = 2$, what is the instantaneous rate of change of f$ at x = 2$?

Accepted Answer

$$f'(2) = 4$$

Question 16

If $$\lim_{x 	o a^-} f(x) = L$$ and $$\lim_{x 	o a^+} f(x) = L$$, what can be said about $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$\lim_{x 	o a} f(x) = L$$

Question 17

A chemical reaction's concentration $$C(t)$$ changes over time. If $$C(2) = 5$$ mol/L and $$C(5) = 11$$ mol/L, what is the average rate of change of concentration between $$t = 2$$ and $$t = 5$$?

Accepted Answer

$$\frac{11 - 5}{5 - 2} = 2$$ mol/L per unit time

Question 18

Suppose $$f(x) is $$not defined at $$x = a$$, but $$\lim_{x 	o a} f(x)$$ exists. Which statement is true?

Accepted Answer

The function has a removable discontinuity at $$x = a$$

Average and Instantaneous Velocity, and Limits in Calculus

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