Cumulative Review Solutions
Chapters 1 4
2. The given function is a polynomial function of degree three. The x-intercepts are 1 and 2. Since 1 is a double root, the graph is tangent to the x-axis at x = 1. The y-intercept is 2. Since the coefficient of th
Chapter 9 Cur ve Sketching
Review of Prerequisite Skills
2. c. t2 2t < 3 t2 2t 3 < 0 (t 3)(t + 1) < 3 Consider t = 3 and t = 1. 3. a. y = x7 430x6 150x3 dy = 7x6 2580x5 450x2 dx dy If x = 10, < 0. dx dy If x = 1000, > 0. dx The curve rises upward in quadr
Topic 8A Motion of a Point
1. Rectilinear (straight line) motion
The function
s f t
The first derivative
represents displacement at a given time t.
v f t
represents the instantaneous velocity or
change of displacement with respect to time.
The second der
Topic 5A: Fundamental Identities
Reciprocal Relations:
If
Unit Circle:
Then
sin csc 1
csc
sec
1
cos
cot
Quotient Relations:
Pythagorean Relation:
x2 + y2 = 1
cos2 sin 2 1
cos2 sin 2 1
cos
2
2
cos2 sin 2 1 sin
1
Practice
Change to an expression conta
Topic 6F Derivative of a Function Raised to a Power (Chain Rule)
To determine which rule to apply, you must correctly identify the function to be solved.
Ex 1)
y (1 2 x)3
- is a power function
- must expand to take derivative
Chain Rule:
y un
where u is a
Topic 4E The Sine Wave as a Function of Time
P
O
A sine curve can be generated by rotating a
vector OP (called a phasor) using a constant
angular velocity (rad/s).
1
Compare the equation
y a sin(bx c) d
With
y a sin( t ) d
Ex 1) Write the equation for a s
Graphing Practice
1. Give the coordinates of the points graphed (each box represents 1 unit):
A:
B:
D:
E:
C:
2. Plot the following points on a rectangular coordinate system.
M(5, 3)
N(0, -3)
P(-4, 5)
Q(5, 0)
R(-4, -6)
S(-4, -3)
3. Give the quadrant in whi
Topic 6A Using motion to illustrate the concept of derivatives
While the topics of algebra, logarithms, and trigonometry are of fundamental importance
to the scientist, a wide variety of technical problems cant be solved using only these
tools.
Many probl
Topic 6B Limits
From the last section, we can develop both average and instantaneous speeds using the
limit process.
In fact, limits are fundamental to all of calculus. In this section, we will give a basic
definition of a limit, and show how it can be ap
Cumulative Review Solutions
Chapters 57
1. c. x2 + 16y2 = 5x + 4y We differentiate both sides of the equation with respect to x: d2 d (x + 16y2) = (5x + 4y) dx dx 2x + 32y (32y 4) dy dy =5+4 dx dx dy = 5 2x dx dy 5 2x = . dx 32y 4 d. 2x2 xy + 2y = 5 We di
Chapter 7 The Logarithmic Function and Logarithms
Review of Prerequisite Skills
4. The increase in population is given by f(x) = 2400(1.06)x f(20) = 2400(1.06)20 = 7697. The population in 20 years is about 7700. The function representing the increase in b
Chapter 6 Exponential Functions
Review of Prerequisite Skills
3. d. 21 + 22 31 11 + 24 = 1 3 11 + 24 1 3
y3
6.
a.
y y1 y x
x
=
12 12 6+3 4 9 4
= = 5.
b. (i) The graph of y1 is vertically compressed by one-half to form the graph of y2. (ii) The graph of y1
Chapter 4 Derivatives
Review of Prerequisite Skills
1. f. 4 p 7 6 p 9 24 p16 = 12 p15 12 p15 = 2p
Exercise 4.1
y y
3.
i.
( 3a )[ 2a ( b) ]
4 3 3
12a b
5
2
6a 1b 3 12a 5b b = 6 2a =
3
1
x 1
x
6.
d.
( x + y)( x y) ( x + y) 5( x y ) 10 ( x + y) 10 = 5 ( x +
Chapter 2 Polynomial Equations and Inequalities
Review of Prerequisite Skills
1. b. 3 ( x 2) + 7 = 3 ( x 7) 3x 6 + 7 = 3x 21 3x + 1 = 3x 21 0 x = 22 There is no solution. 2. c. 4 x 5 2( x 7) 4 x 5 2 x 14 2 x 9 9 x 2 or x 4.5 -6 -5 d. 4 x + 7 < 9 x + 17 5x
Chapter 1 Polynomial Functions
Review of Prerequisite Skills
2. g.
( x + n) 2 9 = ( x + n + 3)( x + n 3)
2
c. y 5 y 4 + y 3 y 2 + y 1 = y 4 ( y 1) + y 2 ( y 1) + ( y 1) = ( y 1) ( y 4 + y 2 +1)
h. 49u 2 ( x y )
= ( 7u + x y )( 7u x + y ) x 16
4
e. 9 ( x +
Appendix A
Exercise
4. 2x b. cos = = and is an angle in the third 3r quadrant. Since x2 + y2 = r2, 4 + y2 = 9 y = 5. Hence, sin = 5 5 and tan = . 3 2 d. c. sin4x cos4x = 1 2cos2 x L.S. = sin4x cos4x = (sin2x + cos2x)(sin2x cos2x) = (1)(1 cos2x cos2x) = 1
Appendix B
Exercise B1
2. a. The general antiderivative of f(x) = 12x2 24x + 1 1 1 is F(x) = 12 x3 24 x2 + x + C 3 2 = 4x3 12x2 + x + C. Since F(1) = 2, we have: F(1) = 4 12 + 1 + C = 2. Thus, 7 + C = 2 C = 5. The specific antiderivative is F(x) = 4x3 12x
Read the Journal Article Historical Reflections On Teaching Trigonometry that is posted on my website
under Unit 6 and also on the T Drive. There are only six pages so make sure you read the article for
understanding. As you read, take Cornell notes and w