3.1.3 The Derivative

Question 1

The Derivative

Answer 1

• Tangent lines are graphic representations of instantaneous rates of change.
• To find the slope of a tangent line, take the limit as the change in the independent variable approaches zero.
• The derivative is a function that gives you the instantaneous rate and slope of the tangent line at a point. The derivative got its name from the fact that it is derived from another function.

Question 2

note

Answer 2

• One way to approximate the instantaneous rate of change at a point is to calculate the average rate of change between that point and another point nearby. The closer the second point is, the better the approximation will be.
• As the distance between the two points diminishes to zero, the line becomes tangent to the curve. The slope of that line is the instantaneous rate of change of the function.
• Here is a systematic approach for finding the instantaneous rate of change of a function fat a point (x, f(x)).
• First, consider a nearby point whose x-coordinate is off by a small amount, x.Its coordinates will be (x+ x, f(x+ x)). Second, express the average rate of change between the two points.
• Finally, use a limit to reduce the offset to zero.
• The limit is called the derivative of fat x and is denoted by a prime symbol “ ‘ ”. For a specific value of x, the derivative produces the slope of the line tangent to the function at that point. More generally, the expression defines the derivative function, which takes an x-value as its input and produces the slope of the corresponding tangent line.

Question 3

Given that f(t)=2t^2−4t and that f′(t)=4t−4, find the instantaneous rate of change at t=3.

Answer 3

f′(3)=8

Question 4

Consider the function f(x)=e^x.
Suppose you are given that f’(x)=e^x.
What can you conclude about the limit
lim Y→0 e^Y−1 / Y?

Answer 4

lim Y→0 e^Y−1 / Y=1

Question 5

Suppose f(x)=−2x^2. What is the instantaneous rate of change of f(x) when x=4?

Answer 5

-16

Question 6

Suppose f(x)=x^2−3. What is the slope of the line tangent to f(x) at x=3?

Answer 6

6

Question 7

Which of the following is the correct definition of the derivative?

Answer 7

f’(x) = lim Δx->0 f(x+Δx) -f(x) / Δx

Question 8

Suppose f(x)=3x^2. What is f′(2)?

Answer 8

12

Question 9

The temperature begins to drop on a winter day in Maine as a cold front moves in. The temperature is given by the function T(x)=12−1/12x^5, where x is the time in hours and T(x) is the temperature in Fahrenheit. Given that T′(x)=−512x4, when will the temperature be falling at a rate of 5 degrees per hour?

Answer 9

1.9 hours

Question 10

If you can’t find instantaneous rates of change by using algebra because the denominator will equal zero, why can you take the derivative of a function?

Answer 10

Because you never actually divide by zero. By taking the limit you get around the problem by finding the value arbitrarily close to the point.

Question 11

Suppose you are given the function f(x)=x^3−x+5 and its derivative f′(x)=3x^2−1.What is the slope of the line tangent to the graph of f(x) at x=−2?

Answer 11

11

Question 12

Using the definition of the derivative, calculate the derivative of the function g (x) = −4x + 2.

Answer 12

g′(x)=−4

Question 13

Given that f(t)=3t^2+1, find the instantaneous rate of change at t=2.

Answer 13

f′(2)=12