Name_________________________________________
AP AB Calculus
Chapter 3B
Notepacket
Date/Day
11/12 – day 1
11/13 – day 2
11/16 – day 3
11/17 – day 4
11/18 – day 5
11/19 – day 6
Topic
2nd Derivative Test / Optimization
Linear Approximation
Motion
Review
Review
Test
Assignment
Worksheet 1
Worksheet 2
Worksheet 3
Worksheet 4
AP Calculus AB
Optimization
Notes 3.7 – Optimization Day 1
Learning Targets: I can apply the second derivative test to find relative extrema of a function.
Second Derivative Test :
If c is a critical number of f ( x ) , and



f ''(c)  0 , then there is a relative minimum at c.
f ''(c)  0 , then there is a relative maximum at c.
f ''(c)  0 , then there is no conclusion possible, and you must revert back to the first derivative test.
*Critical numbers found from f ' not defined cannot be used in the second derivative test, but can still
yield extrema.
To use second derivative test:
1. Set f '  0 to find all critical points.
2. Plug critical points into second derivative.
3. If:
f ''(c)  0 , then there is a relative minimum at c.
f ''(c)  0 , then there is a relative maximum at c.
f ''(c)  0 , then there is no conclusion possible, and you must revert back to the first derivative
test.
Use the second derivative test to find the x-coordinates of all relative extrema.
Ex1.
f ( x)  x 3  2 x 2  4 x  3
Ex3. f ( x)  x 4
Ex2.
g ( x)  x 3  5 x 2  7 x
I can solve applied minimum and maximum (optimization) problems.

I can write an equation or set of equations to represent a given scenario.

I can apply the derivative tests to optimize a given scenario.
We are often asked to solve problems that use phases such as “greatest profit”, “least cost”, “least amount of time”,
“greatest distance”, “optimum size” etc. These problems are called Optimization problems. Optimization problems
determine minimum or maximum values.
Guidelines for Solving Applied Minimum and Maximum Problems (p. 219)
1. Identify all given quantities and all quantities to be determined. If possible make a sketch.
2. Write a primary equation for the quantity that is to be maximized or minimized (a review of several useful
formulas from geometry is presented in the back cover of the book.)
3. Reduce the primary equation to one having a single independent variable. This may involve the use of
secondary equations relating the independent variables of the primary equation.
4. Determine the feasible domain of the primary equation. That is determining the values for which the stated
problem makes sense.
5. Determine the desired maximum or minimum value by the calculus techniques discussed in Sections 3.1-3.4
(finding critical numbers).
Examples:
1. A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions
should be used to maximize the enclosed area.
2. An open box is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from
each corner and turning up the sides. What are the dimensions of the box that has the greatest volume?
3. A rectangle has its base on the x-axis and its upper two vertices on the parabola y  12  x 2 . What is the largest
area the rectangle can have?


1
2
4. Find the point on the graph of the function f ( x)  x 2 closest to the point  2,  .
5. A man is in a boat that is 2 miles from the nearest point on the shore. He is going to point Q (as seen in the
diagram below) which is 3 miles down the coast and 1 mile inland. He can row at 2 miles per hour, and walk at 4
miles per hour. Toward what point on the coast should he row in order to reach point Q in the least amount of
time?
Boat
2
1
Q
If you need extra practice, please go online and:
watch the Khan Academy video at
http://www.khanacademy.org/math/calculus/differential-calculus/v/optimization-with-calculus-1
watch the Khan Academy video at
http://www.khanacademy.org/math/calculus/differential-calculus/v/optimization-with-calculus-2
AP Calculus AB
Notes - Tangent Line Approximation
Learning Target: I can use the tangent line to a graph to approximate a value.

I can write an equation of a tangent line.

I can approximate a value of a curve using a tangent line.

I can determine if an approximation is an over- or under-approximation using concavity.
Tangent Line Approximation (or Linear Approximation)
If you take the equation of the tangent line: y  y1  m  x  x1  and let x  a .
An equation of the tangent line to a curve at the point (a, f (a)) is:
y  f (a)  f '(a)  x  a 
y  f (a)  f '(a)  x  a 
So,
providing that f is differentiable at a.
Since the curve of f ( x ) and the tangent line are close to each other for points near x  a , f ( x)  f (a)  f '(a)  x  a  .
Example 1.
Write an equation of the tangent line to f  x   x3 at (2, 8). Use the tangent line to find the approximate values of
f 1.9 and f  2.01 .
Examine the graph. Is the approximation an under- or over-approximation? Why? What conclusions can you make
regarding concavity and tangent line approximation?
Example 2.
If f is a differentiable function and f (2)=6 and f '  2   
1
, find the approximate value of f (2.1).
2
Example 3.
The slope of a function at any point (x, y) is 
x 1
. The point (3, 2) is on the graph of f. (a) Write an equation of the line
y
tangent to the graph of f at x =3. (b) Use the tangent line in part (a) to approximate f (3.1).
AP Calculus AB
Notes – Particle Motion
Name
Date
Period
Position Function

The position of an object x  t  is a function of time, t. The object moves on a line toward or away from
a fixed reference point (e.g., the origin).
Velocity Function

The velocity of the object is the derivative of the position: v  t   x '  t  .


Velocity has direction as indicated by its sign and a magnitude called speed. Thus, velocity is a vector.
Positive velocity indicates motion in the positive direction; negative velocity indicates motion in the
negative direction.
A moving object is “at rest” when its velocity is (momentarily) zero.
The units of velocity are distance divided by time (miles/hour, feet per second, etc.)
Think of velocity as pulling the particle in the direction it is moving.



Acceleration

The acceleration of the object is the derivative of the velocity and the second derivative of the position:
a  t   v '  t   x '' t  .

Acceleration is also a vector. Its sign indicates the direction in which the velocity is changing: positive
acceleration indicates the velocity is increasing; negative decreasing.
The units of acceleration are velocity divided by time ((miles/hour)/hour, (feet per second)/per second,
etc.)
Think of acceleration as pulling the velocity one way or the other (increasing or decreasing it).


Speed

The speed of the object is the absolute value of the velocity: Speed = v(t ) .




Speed is the length of the velocity vector.
Speed is a number, not a vector.
The units of speed are the same as the units of velocity (miles/hour, feet per second, etc.)
Speed is increasing when the velocity and acceleration act in the same direction – they have the same
sign; speed is decreasing when the velocity and acceleration act in different direction – they have
different signs.
If the velocity graph is moving away from the t-axis, the speed is increasing; if the velocity graph is
moving towards the t-axis, the speed is decreasing.

Given the following velocity graphs, determine the sign of the velocity and acceleration on the given intervals and then
whether the speed is increasing or decreasing.
Velocity:
Acceleration:
Speed:
Velocity:
Acceleration:
Speed:
Velocity:
Acceleration:
Speed:
Velocity:
Acceleration:
Speed:
Would the same conclusions hold if the graphs were linear?
Examples:
1. The distance of a particle from its initial position is given by s  t   t  5 
9
, where s is feet and t is minutes. Find
t 1
the velocity at t  1 minute in feet per minute.
A. 
5
4
B.
13
4
C.
1
2
D. 
9
4
E. 
7
4
2. The distance of a particle from its initial position is measured every 5 seconds and provided in the table below. Use
the data to answer the questions that follow.
Time (sec)
0
5
10
15
20
Distance (ft)
0
7
17
25
30
A. Estimate the velocity of the particle at t  15 seconds. Include units.
B. What is the average velocity on the time interval 5, 20 seconds? Include units.
3. The graph of the velocity of a particle is given below. On the same axes, draw the acceleration graph and a sketch of
the position graph. Label graphs.
4. The number of liters of water remaining in a tank t minutes after the tank has started to drain is
R  t   2t 3  20t 2  72t  820 . At what moment is the water draining the fastest?
A. 0 min
B. 2 min
1
3
C. 3 min
1
3
D. 10 min
E. It drains at the same
rate the whole time.
  . For
5. (Calculator Active) A particle moves along a horizontal line with velocity function given by v  t   cos ln t
2
t  1, t  2, and t  3, at which times is the particle speeding up?
A. t  1 only
B. t  2 only
C. t  3 only
D. t  1 and 2 only
E. t  2 and 3 only
AP Calculus AB
Optimization
In Class Optimization Extra Practice
Name______________
Examples:
1.
Find the length and width of a rectangle with perimeter 60 meters and maximum area.
2.
Find the area of the largest rectangle that can be created inside a circle of radius 3 feet.
3.
The sum of the perimeters of two squares is 24. Find the dimensions of the squares that produce the minimum
total area.
4. Find two positive numbers whose product is 185 and whose sum is a minimum.