$$v = \lim_{\Delta t \to 0} \frac{f(t+\Delta t) - f(t)}{\Delta t}$$

Question 1

Which of the following best describes the mathematical definition of instantaneous velocity?

Accepted Answer

$$v = \lim_{\Delta t 	o 0} \frac{f(t+\Delta t) - f(t)}{\Delta t}$$

Question 2

Given $$f(t) = at^n$$, what is the general formula for $$\frac{df}{dt}$$?

Accepted Answer

$$\frac{df}{dt} = $$nat^{n-1}$$

Question 3

If $$\vec{v} = \frac{d\vec{r}}{dt}$$ and $$\vec{a} = \frac{d\vec{v}}{dt}$$, what does $$\vec{a}$$ represent physically?

Accepted Answer

The rate of change of velocity with respect to time

Question 4

Which rule allows you to differentiate a composite function $$f(g(t))$$?

Accepted Answer

Chain rule

Question 5

Using the chain rule, what is $$\frac{d}{dt} f(g(t))$$?

Accepted Answer

$$\frac{df}{dg} \cdot \frac{dg}{dt}$$

Question 6

Suppose $$f(t) = t^2$ and g(t$$) = \sin(t)$$. What is $$\frac{d}{dt} f(g(t))$$?

Accepted Answer

$$2\sin(t)\cos(t)$$

Question 7

If $$\vec{v} is $$constant, what is the area under the velocity-time graph from $$t=a to$$ $$t=b$$?

Accepted Answer

$$v(b-a)$$

Question 8

Which of the following best describes the relationship between integration and differentiation?

Accepted Answer

Integration is the inverse operation of differentiation

Question 9

Given $$F(x) = \int f(x) dx$$, what is $$\frac{dF}{dx}$$?

Accepted Answer

$$f(x)$$

Question 10

If $$\frac{d}{dt} \left( \int f(t) dt \right) = f(t)$$, what does this imply about the operations of integration and differentiation?

Accepted Answer

They are inverse operations

Question 11

A car moves such that its position is given by $$x(t) = 3t^2$ + 2t + 1. $$What is its instantaneous velocity at $$t=2 s$$?

Accepted Answer

$$v = 14 m/s$$

Question 12

If $$\vec{a} = \frac{d\vec{v}}{dt}$$ and $$\vec{v} = \frac{d\vec{r}}{dt}$$, what is $$\vec{a} in $$terms of $$\vec{r}$$?

Accepted Answer

$$\vec{a} = \frac{d^2$\vec{r}}{dt^2$}$$

Question 13

Which of the following is a correct application of the chain rule for $$\frac{d}{dt} f(g(t))$$?

Accepted Answer

$$\frac{df}{dg} \cdot \frac{dg}{dt}$$

Question 14

If $$f(x) = x^3$, what is $$\int f(x) dx$$?

Accepted Answer

$$\frac{1}{4}x^4$ + C$$

Question 15

A ball is thrown vertically upward with velocity $$v_0. $$Which integral gives the displacement after time $$t$$?

Accepted Answer

$$\$$int_0$$^t (v_0$ - gt) dt$$

Calculus Concepts in Physics: Motion, Differentiation, and Integration

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