Question 1

Given $$f(t) = t^2$ represents the distance in cm of a rat from the origin after t$ seconds, what is the average speed of the rat from t = 1$ to t = 3$ seconds?

Accepted Answer

$$4 cm/sec$$

Question 2

If $$f(t) = t^2$, what is the instantaneous speed of the rat at t = 5$ seconds?

Accepted Answer

$$10 cm/sec$$

Question 3

What is the average rate of change of temperature from $$t = 6 to$$ $$t = 8 if$$ $$f(t) = t^2$?

Accepted Answer

$$14$$ degrees/hour

Question 4

Which of the following best describes the difference between speed and velocity?

Accepted Answer

Speed is always non-negative, while velocity can be positive or negative depending on direction.

Question 5

Suppose the rat's position is given by $$f(t) = t^2$. How many centimeters has the rat run by the time its speed is 8$ cm/sec?

Accepted Answer

$$16 cm$$

Question 6

What is the slope of the tangent line to $$f(t) = t^2$ at t = 3$?

Accepted Answer

$$6$$

Question 7

If the rat runs along the $$x-$$axis and its position is $$f(t) = t^2$, what is the average speed from t = 2$ to t = 4$?

Accepted Answer

$$6 cm/sec$$

Question 8

Which of the following statements about the derivative is correct?

Accepted Answer

The derivative gives the instantaneous rate of change of a function at a point.

Question 9

A rat runs along the $$x-$$axis. After $$t$$ seconds, its position is $$f(t) = t^2$. What is the average speed from t = 1$ to t = 1.1$?

Accepted Answer

$$2.1 cm/sec$$

Question 10

Which of the following best describes the tangent line problem?

Accepted Answer

Finding the slope of the tangent line at a point by taking the limit of the slopes of secant lines.

Question 11

If $$f(t) = t^2$, what is f$$'(t)$$?

Accepted Answer

$$2t$$

Question 12

Suppose the temperature at time $$t is $$given by $$f(t) = t^2$. What is the instantaneous rate of change of temperature at t = 7$?

Accepted Answer

$$14$$ degrees/hour

Question 13

Which of the following is a correct interpretation of the derivative in a real-world context?

Accepted Answer

It represents how fast something is changing at a specific moment.

Question 14

If the rat's speed is given by $$f'(t) = 2t$$, what is the rat's speed at $$t = 0$$?

Accepted Answer

$$0 cm/sec$$

Question 15

Which of the following best explains why the instantaneous rate of change is defined as a limit?

Accepted Answer

Because it is the limit of the average rate of change as the interval approaches zero.

Lecture 15: Introduction to Derivatives and Rates of Change

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