AP Calculus 30

Supplemental Material

Unit 5 S5.1 Curve Sketching

S5.2 Optimization
S5.3 More Optimization

Unit 6 S6.1 Integration by sight

S6.2 Integration by substitution
S6.3 Fundamental Theorem
S6.4 Definite Integration

Unit 7 S7.1 Riemann Sums

S7.2 Slope Fields

Unit 8 S8.1 Area Above or Below a Curve

S8.2 Area Between Two Curves
S8.3 Volume
Unit 5 Supplemental Material

S5.1 Curve Sketching

For the following examples find:

A) Sign analysis of f’(x) and f’’(x)
B) Open intervals in which f is increasing or decreasing
C) Open intervals in which f is concave up or down.
D) Critical numbers and all relative extrema.
E) All inflection points.
F) X-y intercepts
G) Equations of asymptotes
H) A careful sketch of the graph that illustrates each of the features found
above.

Example: 1. f(x)=x4-4x3+16x
 2
x +1
2.f(x)=
x
Assignment pg. 259 #1-13 odd.
S5.2 Optimization
 Helpful Strategy
1. Problem – Understand what the word problem is asking you
to find. Will usually be maximize or minimize a particular quantity.
2. Diagram – Many optimization problems require a diagram,
clearly labeled using variables to identify unknown dimensions.
3. Function – Construct an equation in terms of other
variables. Use relationships among the variables to otain Q in terms of a
single variable. (Usually use Pythagoras theorem or ratio’s of corresponding
sides in similar triangles).
4. Domain – determine the domain of the function by carefully
reading the problem.
5. Global Maximum/Minimum – assuming the domain is
closed:
a) find the derivative.
b) determine critical numbers.
c) evaluate function at critical numbers.
d) evaluate function at the end points.
e) the greatest is the maximum, while the least is the
minimum.
6. Conclusion – Be sure to answer the question originally
asked.

Example1: Two numbers add to 12. Find them if twice one of them
plus their product is a maximum.

Example 2: A rectangular sheet of metal, 8 cm wide and 100 cm long,

is to be folded along the center of its length to form a triangular trough. In
order to maximize the carrying capacity of the trough, what should the width
of the trough be at the top? You may assume tat the carrying capacity is
maximized if the cross sectional area is maximized.

Assignment pg 277 # 2,4,8,13,15,19.

S5.3 More Optimization
1. We have a piece of cardboard that is 50 cm by 20 cm and we are
going to cut out the corners and fold up the sides to form a box.
Determine the height of the box that will give a maximum volume.
 2. We are going to fence in a rectangular field. If we look at the field
from above the cost of the vertical sides are $10/ft, the cost of the
bottom is $2/ft and the cost of the top is $7/ft. If we have $700
determine the dimensions of the field that will maximize the
enclosed area.

Link on classroom in Unit 5 section.

S6.1 Integration by sight
The reverse process of finding a derivative or differential is known as integration
or as finding the indefinite integral.

Use trial and error to see how you do on these first few examples:

1. Find a function f(x) whose derivative, f’(x), is 3x2 + 4x.

2. Find a function whose differential is cos x + x8+x-1+12.

1
3. Find a function f(x) such that f’(x)= 2 dx.
1+ x

To denote the operation of integration we use what is called an integral sign,

sigma (sum) notation , ∫ , in front of the expression. The symbol ∫f(x) dx means:

1. What function has f(x) as its derivative when this function is

differentiated with respect to x?
2. What must I differentiate relative to x in order to get the function f(x)?
3. What function has f(x) dx as its differential?
 The process of finding the integral (anti – derivative) is very easy for cases in
which we have a formula for the derivative of the function . Although differentiation of
most functions is relatively routine integration of many functions requires a great deal of
patience, foresight, and skill. You should never get the wrong answer with integration,
because it is easy to check your answer with differentiation. Never a wrong answer but
many times no answer.

Examples: In each of the following examples, integrate by sight

4. ∫x dx

5. ∫8x3dx

dx
6. ∫ 4
x

1
7. ∫ x dx

8. ∫(x+1)(x+2)
 9. ∫12cos2xdx

2
10. ∫ 3−x
x
dx

−1
11. ∫ dx
√ 1−x 2

Assignment pg. 356 # 10 – 53

S6.2 Integration by substitution
Generally most integrals cannot be done by sight. Many integrals that appear
complex can be made much simpler by transforming them into equivalent integrals
through a process known as u-substitution.
 Rule of Thumb: u’s and du’s should never appear on the same side of the =
sign along with the x’s and dx’s.

Examples: Evaluate
1. ∫(x+2)400dx

5
2. ∫ ( 3 x−7 ) 3 dx
3. ∫ cos 5 xsinxdx

5
4. ∫ ( x +3 )
2 4
x dx

2 x +5
5. ∫ x 2+5 x +7 dx
 ∫ e x −4 x ( 6 x 2−8 )dx
3

6.

Assignment pg. 366 #7 – 61 odd.

S6.3 Fundamental Theorem

This theorem provides the tools for evaluating an infinite sum by means of a
definite integral.
Riemann Sums: The notation and formulas for finding the Right and
Left Riemann Sums (also known as the upper and lower sums), as
well as the Midpoint Rule and the Trapezoidal Rule.
All of these techniques help us to create the building blocks for our
Integration rules and properties and allows us to formally define
the Definite Integral, which is the infinite limit of our summations.

https://youtu.be/v6Eb59X1BW4

S6.4 Definite Integration

Suppose that f(x) is a continuous function on an open interval I, and suppose that
∫f(x)dx=F(x) + c. Then for any two points a, b in I the number F(b) – F(a) is called the
b

definite integral of f(x) from a to be. We use the notation ∫ f (x )dx = F(b) – F(a). The
a
numbers b and a are called the upper and lower limits of integration respectively, and a
b
must be less than b. Very often we write this F( x)|a in place of F(b) – F(a).

Examples: Evaluate each of the following definite integrals.

1. ∫3x2dx

π
2
2. ∫ cosxdx
0

3. ∫ 1x dx
2
Assignment pg. 370 # 1-35 odd

S7.1 Riemann Sums

An equation involving a derivative is called a differential equation. The order of a

differential equation is the order of the highest derivative involved in the equation.
Example:
1. Solving a Differential Equation
 dy 2
Find all functions “y” that satisfy =sec x+ 2 x +5.
dx

2. Solving an Initial Value problem

dy x 2
Find the particular solution to the equation =e −6 x whose graph passes through the
dx
point (1.0).

3. Handling Discontinuity in an Initial Value Problem

dy 2
Find the particular solution to the equation =2 x −sec x whose graph passes through
dx
the point (0,3).

4. Using FTC to solve an Initial Value Problem

2
Find the solution to the differential equation e−t for which f(7) = 3
 dy
5. Graph the family of functions that solve the differential equation =cos x .
dx

S7.2 Slope Fields

Suppose we want to produce example 5 without actually solving the differential equation
dy
=cos x . Since the differential equation gives the slope at any point (x,y) we can draw
dx
a small piece of linearization at that point which approximates the solution curve that
passes through the point. Repeating this process at many points yields an approximation
called a slope field.

Example:
1. Constructing a slope field

dy
Construct a slope field for the differential equation =cos x .
dx
 **If your calculator has a differential equation mode for the graphing slope fields the
usual “Y=” turns into {dy} over {dx} screen and you can enter the function into the
calculator then hit graph.**
Example:
2. Construct a Slope Field for a Non exact Differential Equation

dy
Use a calculator to construct a slope field for the differential equation =x+ y and
dx
sketch a graph of the particular solution that passes through the point (2,0).

3. Matching slope fields with Differential Equations

S8.1 Area Above or Below a Curve

Our study of mathematics in previous courses has given us formulas

for finding area of all different shapes. What formula would you use to find
the area of the shaded area in the following diagram?

Fundamental Theorem of Calculus:

Suppose f(x) is a continuous function defined on the closed interval [a,b]
and f(x)≥0 for all points in the interval. Then the area bounded by the
x-axis, the vertical lines x=a and x=b and the function y=f(x) is given by
b

∫ f ( x ) dx
a

Example: 1. Find the area bounded by the x-axis, the lines x= -2 and
x= 1, and the curve f(x)= 3x2 +2

2. Find the area of the region bounded by the x-axis, the line
−π
x= 4 and x=π, and the curve f(x) = sin 2x + 4
 3. Find the area bounded by the x-axis, the lines x=0 and x=3 and the
1
curve f(x)=
( x−2 )2

Assignment 381 # 1 – 8 by hand

9 – 12 by calculator
Area above a Curve

The area of the region bounded by the continuous function y = f(x)

below, the x-axis above and the vertical lines x = a and x = b is given by

∫−f ( x ) dx
a

Example: 1. Find the area of the region bounded by the function

f ( x )=x −4 x−5 , the x-axis, and the lines x=1 and x=4.
2
2. Find the total area enclosed by the x – axis the function
f(x) =x2-3x, and the lines x=1 and x=4.

Assignment pg. 381 # 1 – 21

S8.2 Area Between Two Curves

Theorem: If f(x) and g(x) are two continuous functions on the closed
interval [a,b] and f(x)≥ g(x) for all values in the interval, then the area of the
region formed by vertical lines x=a and x=b is given by:
b

∫ [ f ( x )−g(x) ] dx
a
Example: 1. Find the area bounded by f(x)=x2-2x+3, g(x)=-1-x2, and the
lines x=-1 and x=2.

2. Find the area bounded by the curves y=x3 and y=x2+6x.

Assignment pg 384 # 1 – 18
S8.3 Volume
Example:

1. Find the volume of the solid that is generated when the region under
y=2x from 0 to 2 is rotated about the x-axis.

2. Find the Volume of the solid that is generated when the region under
y=x3 from 1 to 2 is rotated about the x-axis. Include a sketch of the
region that is to be rotated and a sketch of the solid.
3. Find the Volume of the solid that is generated when the region
π
between y=sin x and y=cos x from 0 to 4 is rotated about the x-axis.