1. Introduction to Calculus
Precalculus
Sixth Edition
Chapter 11
Introduction to Calculus
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2. 11.4 Introduction to Derivatives

3. Learning Objectives Find slopes and equations of tangent lines.
Find the derivative of a function.
Find average and instantaneous rates of change.
Find instantaneous velocity.

4. Slopes and Equations of Tangent Lines (1 of 2)
The average rate of change between any two points on the graph of a function is the slope of the line containing the two points. We call this line the secant line.

5. Slopes and Equations of Tangent Lines (2 of 2)
The figure shows how the secant line between points P and Q changes as h approaches 0. The limiting position of the secant line is called the tangent line to the graph of f at the point P = (a, f(a)).

6. Slope of the Tangent Line to the Curve at a Point
The slope of the tangent line to the graph of a function y = f(x) at (a, f(a)) is given by
provided that this limit exists. This limit also describes
the slope of the graph of f at (a, f(a)).
the instantaneous rate of change of f with respect to x at a.

7. Example 1: Finding the Slope of a Tangent Line
Find the slope of the tangent line to the graph of
at (4, 12). At the point (4, 12), a = 4.
The slope of the tangent line to the graph of f at (4, 12) is 7.

8. Example 2: Finding the Slope-Intercept Equation of a Tangent Line (1 of 2)
Find the slope-intercept equation of the tangent line to the graph of
at (1, 1).
We begin by finding the slope of the tangent line to the graph at (1, 1).

9. Example 2: Finding the Slope-Intercept Equation of a Tangent Line (2 of 2)
Find the slope-intercept equation of the tangent line to the graph of
at (1, 1).
The tangent line is given to pass through (1, 1). We found the slope of the tangent line to be
The slope-intercept equation of the tangent line to the graph is

10. The Derivative of a Function
Let y = f(x) denote a function f. The derivative of f at x, denoted by
read “f prime of x,” is defined by
provided that this limit exists. The derivative of a function f gives the slope of f for any value of x in the domain of

11. Example 3: Finding the Derivative of a Function (1 of 2)
Find the derivative of
at x. That is, find

12. Example 3: Finding the Derivative of a Function (2 of 2)
Find the slope of the tangent line to the graph of
at x = −1 and at x = 3.
The slope of the tangent line at x = −1 is −7.
The slope of the tangent line at x = 3 is 1.

13. Average and Instantaneous Rates of Change
Average Rate of Change The average rate of change of f from x = a to x = a + h is given by the difference quotient
Instantaneous Rate of Change The instantaneous rate of change of f with respect to x at a is the derivative of f at a:

14. Example 4: Finding Average and Instantaneous Rates of Change (1 of 4)
The function
describes the volume of a cube, f(x),
in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the average rate of change of the volume with respect to x as x changes from 4 inches to 4.1 inches.
The average rate of change in the volume is cubic inches per inch as x changes from 4 to 4.1 inches.

15. Example 4: Finding Average and Instantaneous Rates of Change (2 of 4)
The function
describes the volume of a cube, f(x),
in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the average rate of change of the volume with respect to x as x changes from 4 inches to 4.01 inches.
The average rate of change in the volume is cubic inches per inch as x changes from 4 to 4.01 inches.

16. Example 4: Finding Average and Instantaneous Rates of Change (3 of 4)
The function
describes the volume of a cube, f(x),
in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the instantaneous rate of change of the volume with respect to x at the moment when x = 4 inches.

17. Example 4: Finding Average and Instantaneous Rates of Change (4 of 4)
The function
describes the volume of a cube, f(x),
in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the instantaneous rate of change of the volume with respect to x at the moment when x = 4 inches.
The instantaneous rate of change of f is the derivative. To find the instantaneous change of f at 4, find
The instantaneous rate of change of the volume at the moment when x = 4 inches is 48 cubic inches per inch.

18. Instantaneous Velocity
Suppose that a function expresses an object’s position, s(t), in terms of time, t. The instantaneous velocity of the object at time t = a is
Instantaneous velocity at time a is also called velocity at time a.

19. Example 5: Finding Instantaneous Velocity (1 of 6)
A ball is thrown straight up from ground level with an initial velocity of 96 feet per second. The function
describes the ball’s height above the ground, s(t), in feet, t seconds after it is thrown. What is the instantaneous velocity of the ball after 4 seconds? Instantaneous velocity is given by the derivative of a function that expresses an object’s position, s(t), in terms of time, t. The instantaneous velocity of the ball at 4 seconds is

20. Example 5: Finding Instantaneous Velocity (2 of 6)
The instantaneous velocity of the ball at 4 seconds is

21. Example 5: Finding Instantaneous Velocity (3 of 6)
A ball is thrown straight up from ground level with an initial velocity of 96 feet per second. The function
describes the ball’s height above the ground, s(t), in feet, t seconds after it is thrown. What is the instantaneous velocity of the ball after 4 seconds? The instantaneous velocity of the ball at 4 seconds is
The instantaneous velocity of the ball at 4 seconds is −32 ft/sec.

22. Example 5: Finding Instantaneous Velocity (4 of 6)
A ball is thrown straight up from ground level with an initial velocity of 96 feet per second. The function
describes the ball’s height above the ground, s(t), in feet, t seconds after it is thrown. What is the instantaneous velocity of the ball when it hits the ground? The ball hits the ground when its s(t), its height above the ground, is 0.

23. Example 5: Finding Instantaneous Velocity (5 of 6)
The instantaneous velocity of the ball when it hits the ground is

24. Example 5: Finding Instantaneous Velocity (6 of 6)
A ball is thrown straight up from ground level with an initial velocity of 96 feet per second. The function
describes the ball’s height above the ground, s(t), in feet, t seconds after it is thrown. What is the instantaneous velocity of the ball when it hits the ground? The instantaneous velocity of the ball when it hits the ground is
The instantaneous velocity of the ball when it hits the ground is −96 ft/sec.