the derivative. objectives students will be able to use the “newton’s quotient and limits”...
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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.