Table of Contents
ix ix
Chapter 7: Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
The Problems You’ll Work On ..................................................................... 59
What to Watch Out For ................................................................................. 59
Using Implicit Differentiation to Find a Derivative .................................... 60
Using Implicit Differentiation to Find a Second Derivative ...................... 60
Finding Equations of Tangent Lines Using Implicit
Differentiation............................................................................................. 61
Chapter 8: Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . .63
The Problems You’ll Work On ..................................................................... 63
What to Watch Out For ................................................................................. 63
Finding and Evaluating Differentials ........................................................... 64
Finding Linearizations ................................................................................... 64
Using Linearizations to Estimate Values .................................................... 64
Understanding Related Rates ....................................................................... 64
Finding Maxima and Minima from Graphs ................................................. 66
Using the Closed Interval Method ............................................................... 67
Finding Intervals of Increase and Decrease ............................................... 68
Using the First Derivative Test to Find Local Maxima
and Minima ................................................................................................. 68
Determining Concavity ................................................................................. 68
Identifying Inflection Points ......................................................................... 69
Using the Second Derivative Test to Find Local Maxima
and Minima ................................................................................................. 69
Applying Rolle’s Theorem ............................................................................ 69
Using the Mean Value Theorem ................................................................... 70
Applying the Mean Value Theorem to Solve Problems ............................ 70
Relating Velocity and Position ..................................................................... 70
Finding Velocity and Speed .......................................................................... 70
Solving Optimization Problems ................................................................... 71
Doing Approximations Using Newton’s Method ....................................... 73
Approximating Roots Using Newton’s Method .......................................... 73
Chapter 9: Areas and Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
The Problems You’ll Work On ..................................................................... 75
What to Watch Out For ................................................................................. 75
Calculating Riemann Sums Using Left Endpoints ...................................... 76
Calculating Riemann Sums Using Right Endpoints ................................... 76
Calculating Riemann Sums Using Midpoints .............................................. 77
Using Limits and Riemann Sums to Find Expressions
for Definite Integrals .................................................................................. 77
Finding a Definite Integral from the Limit
and Riemann Sum Form ............................................................................ 78
Using Limits and Riemann Sums to Evaluate
Definite Integrals ........................................................................................ 78