The Derivative

Objectives

Students will be able to• Use the “Newton’s Quotient and limits”

process to calculate the derivative of a function.

• Determine the slope of the tangent line to a given curve when shown a graph.

• Find the equation of the secant line through two points on a function.

• Find the equation of the tangent line to a given function through a specified point.

• Solve problems involving the derivative.

Newton’s Quotient and limits process for finding the derivative:

• Add h (where ) to a and compute f(a + h).

• Compute the change in the function value .

• Calculate and simplify Newton’s quotient

for .

• Calculate the limit

)()( afhaf

0h

hafhaf )()( 0h

h

afhafh

)()(0

lim

Example 1

Estimate the slope of the tangent line to the curve in the graph to the right.

Example 2

Estimate the slope of the tangent line to the curve in the graph to the right.

Example 3

Estimate the slope of the tangent line to the curve in the graph to the right.

Example 4

Find each of the following for the function below: 276)( xxf

)3(f

)1(f

)2(f

)(xf

Example 5

Find each of the following for the function below:

xxf

5)(

)3(f

)1(f

)2(f

)(xf

Example 6

Find each of the following for the function below:

xxf 3)(

)3(f

)2(f

)0(f

)(xf

Example 7

For the function below, find the equation of the secant line through the points x = 3 and x = 5.

xxxf 2)( 2

Example 8

For the function below, find the equation of the tangent line through the point x = 3.

xxxf 2)( 2

Example 9

For the function below, find the equation of the secant line through the points x = 25 and x = 36.

xxf )(

Example 10

For the function below, find the equation of the tangent line through the point x = 25 .

xxf )(

Example 11

Find the x-values where the function to the right is not differentiable.

Example 12

For the function shown in the sketch to the right, give the intervals (or points) on the x-axis where the rate of change of the function f(x) with respect to x is positive, negative, and zero.

Example 13Identify which graph below represent velocity and which represents distance.

Graph 1 Graph 2

Example 14

The cost in dollars of producing x tacos is ,

for .1800 x10005.100375.0)( 2 xxxC

a. Find the exact cost of producing the 101st taco.

b. Use the marginal cost to approximate the cost of producing the 101st taco.