(9/30/08)
Math 10A. Lecture Examples.
Section 2.5. The second derivative
Example 1
The formula M = 70W3/4 is used to predict the basal metabolic rate (the
metabolic rate at rest), measured in Calories per day, of an animal weighing
W kilograms.(1) The gra
(9/30/08)
Math 10A. Lecture Examples.
Section 1.8. Limits
Figure 1 shows the graph of the function,
Example 1
F(x) =
4 x2
4
2x
for
for
for
x<1
x=1
x > 1.
(a) Calculate F(x) at x = 0.9, 0.99, 0.999, and 0.999, 1.1, 1.01, 1.001 and
1.0001. (b) What is lim F
(9/30/08)
Math 10A. Lecture Examples.
Section 1.5. Trigonometric functions
Find the cosine, sine, and tangent of the angle (psi) in the right triangle
of Figure 1.
Example 1
5
FIGURE 1
3
3
Answer: cos =
34
5
sin =
34
5
tan = 3
The largest ferris wheel o
(9/30/08)
Math 10A. Lecture Examples.
Section 1.4. Logarithmic functions
Solve 2x = 5 for x.
Example 1
Answer: x = log2 (5) or x =
ln(5)
ln(2)
Solve log10 y = 1 for y.
Example 2
1
Answer: y = 10
In chemistry the pH of a solution is dened by pH = log10 [H+
(9/30/08)
Math 10A. Lecture Examples.
Section 1.3. New functions from old
Match the functions (a) y = x4 , (b) y = x5 , and (c) y = x1/4 to their
graphs in Figures 1 through 3.
Example 1
y
y
y
3
1
1
2
1
1
1
2
FIGURE 1
x
1
2 1
2 x
FIGURE 2
FIGURE 3
Answer:
(9/130/08)
Math 10A. Lecture Examples.
Section 1.2. Exponential functions
What is the growth factor from 3 to 12?
Example 1
Answer: The growth factor is 4.
What is the growth factor from 20 to 4?
Example 2
Answer: The growth factor is 1 .
5
The function y
(9/30/08)
Math 10A. Lecture Examples.
Section 1.1. Functions and change
Draw the graph of the function A = r2 that gives the area of a circle of
radius r (centimeters).
Example 1
Answer: Figure A1
A (square centimeters)
A = r2
r2
10
Figure A1
1
r
r
(centi
1. (4 points) The graph of y = cos(t) appears below.
Kg 1
0.5 h .
a g
K, [I
g g;
Ray 5?
lkJ 41% l _L _ a? _,_ J_
n neg 3rr 5n V n 7n
" Vi rr E:  2 7T
[ 4 2 1% 4 4 y 2 4
"3 f
as
_0 5 I; a [if/j
L \ y"
K? /
I: m
1.0 WWW
[
t
On the axes below, sketch
(9/30/08)
Math 10A. Lecture Examples.
Section 3.1. Powers and polynomials
Find formulas for (a)
Example 1
Answer: (a)
Example 2
d 3
d 1
d 1/2
(x ), (b)
(x ), and (c)
(x
).
dx
dx
dx
d 1
d 1/2
d 3
(x ) = 3x2 (b)
(x ) = x2 (c)
(x
)=
dx
dx
dx
Give an equation
(9/30/08)
Math 10A. Lecture Examples.
Section 2.1. How do we measure speed?
Imagine that a pilot is ying a small airplane toward the west from an
airport and that the plane is s(t) = t3 + 30t + 100 miles from the airport t
hours after noon (Figure 1). (a)
(9/30/08)
Math 10A. Lecture Examples.
Sections 2.2 and 2.3. The derivative at a point and as a function
The function A = A(t) whose graph is shown in Figure 1 gives the percentage of alcohol in a persons blood t hours after he has consumed three uid
ounce
Math 10C
Midterm 2 Review
All solutions are found in the Online Text by clicking Link to Text or by following the Read, Study, &
Practice link and choosing the appropriate section.
1. (14.3) Find the equation of the tangent plane at the given point.
1
a.
14.1 SOLUTIONS
951
CHAPTER FOURTEEN
Solutions for Section 14.1
Exercises
1. If h is small, then With h = 0.01, we find fx (3, 2) f (3 + h, 2)  f (3, 2) . h
3.012 (2+1)
f (3.01, 2)  f (3, 2) = fx (3, 2) 0.01 With h = 0.0001, we get fx (3, 2)
0.01
2 2

The Differential for Functions of 2 Variables
Recall that for functions of one variable we defined the differential as the change in f as we
moved along the tangent line letting x change by a small amount dx.
We obtained df = f ' (x)dx.
For functions of 2
1180
37
35 (a) yzplane: circle (y + 3)2 + (z 2)2 = 3
xzplane: none
xyplane: point (1, 3, 0)
z
5
i = 0.15
B
(b) Does not intersect
i = 0.1
1000
i = 0.05
y
x
years
37 (8, 0,
t 13.86
3)
39 y = 1 is a plane, not a line
41 Distance is 5
43 (2, 1, 5)
39 (a)
14.1 SOLUTIONS
951
CHAPTER FOURTEEN
Solutions for Section 14.1
Exercises
1. If h is small, then With h = 0.01, we find fx (3, 2) f (3 + h, 2)  f (3, 2) . h
3.012 (2+1)
f (3.01, 2)  f (3, 2) = fx (3, 2) 0.01 With h = 0.0001, we get fx (3, 2)
0.01
2 2

Find and classify critical points
2
Useful facts: The discriminant = fxx fyy fxy at a critical point P (x0 , y0 ) plays the
following role:
1. If (x0 , y0 ) > 0 and fxx (x0 , y0 ) > 0, then f has a local minimum at (x0 , y0 ).
2. If (x0 , y0 ) > 0 and fxx
Name:
PID #:
Section:
MATH 10A
PRACTICE
FINAL
Instructions
Read each question carefully, and answer each question completely.
Show all of your work. No credit will be given for unsupported answers.
Write your solutions clearly and legibly. No credit wi
Chapter 26: 1, 2, 3, 4, 7, 8, 23, 25, 26, 28, 29, 30, 31, 34, 36, 38, and 40
1)
A. Chisquare test of Independence. 1 sample, 2 vars if account type independent of trade type
B. Some other statistics test  account size is quantative, not counts
C. Chisq
Chapter 10: 2, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 28, and
30.
2)
A. Curved pattern, rexpress to straighten the relationship
B. Fan shape, reexpress to equalize spread
C. No pattern, so fine
4)
A. Wavy pattern, so it cont
Chapter 9: 1, 5, 6, 8, 11, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 27, 29, 31, 32 and 34.
1)
a. There is a clear change in trend over the past 100 years from 1890 1940, the trend is
linear and consistent, but then at 1940, the age dropped suddenly to
Chapter 7: 1, 3, 5, 6, 7, 8, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29,
30, 32, 34, 35, 36, 37, 38, and 42.
A.
a.
B.
a.
C.
a.
D.
a.
1) Association:
Apples: weight in grams, weight in ounces
explanatory, response To predict weight in g
Chapter 6: 4, 5, 9, 12, 17, 18, 19, 21, 23, 24, 27, 29, 31, 33, 34, 36, 37, 39, 41, 43, 44, 46, and
47
4) A specialty foods company sells gourmet hams by mail order. The hams vary in size from
4.15 to 7.45 pounds, with a mean weight of 6 pounds and standa
Chapter 5: 2, 7, 8, 11, 13, 17, 20, 21, 25, 26, 27, 29, 30, 35, 36, 37, 38, 39, and 40.
2) Find an article in a newspaper, magazine, or the Internet that shows a time plot.
A.
B.
C.
D.
Does the article discuss the Ws?
Is the timeplot appropriate for the d
Chapter 4: 1, 3, 5, 7, 8, 9, 11, 12, 15, 17, 18, 19, 20, 32, 34, 35, 36, 39, 44, 45, 46, and 50.
1) Find a histogram that shows the distribution of a variable in a newspaper, a magazine, or the
Internet.
From:
A. Does the article identify the Ws?
B. Discu