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Session 40 FINAL REVIEW
Overview of all material
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Olivia Reicks
Math 160
Professor Hunacek
April 30, 2015
Sketch the graph of each equation:
1.1 Example 1: y = 9 – x2
1.1 Example 3: Find the domain of y = √(4 – x)
1.1 Example 3: Find the vertical and horizontal asymptotes of the rational function
f(x) = (3x2 + 3x – 6)/(2x2 – 2)
1.2 Example 2: Given that bacteria grows continuously at relative growth rate 1.386, and that is
N = N0e1.386t. If we start with 25 bacteria, how many bacteria will be present
(A) in 0.6 hour? (B) in 3.5 hours?
1.3 Example 1: Change each logarithmic form to an equivalent exponential form.
(A) Log5 25 = 2
(B) Log9 3 = ½
1.6 (page 77) What are the properties of logarithmic functions?
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2.1 SI #13:
1. Lim x→1 (x5 + 2lnx + 5ex)
2. Lim x→1 (x – 1)/(x2 – 1)
3. Lim x→∞ (210x – 100)/(x2 + 1)
2.1 Example 8: Lim x→ -1 (x| x + 1|)/(x + 1)
2.2 Example 3: Let p(x) = 2x3 – x2 – 7x + 3. Find the limit of p(x) as x approaches ∞ and as x
approaches -∞.
Asymptotes (SI #8):
Find the vertical asymptotes.
a. #56 f(x) = (x2 + 100)/(x2 – 100)
b. #57 g(x) = (x2 + 3x)/(x2 +2x)
Find the horizontal asymptotes.
c. #53 f(x) = (5x + 4)/(x2 – 3x + 1)
d. #54 g(x) = (3x2 + 2x – 1)/(4x2 – 5x + 3)
e. #55 h(x) = x2 + 4/(100x + 1)
2.3 Example 4: Find increasing and decreasing intervals (x + 1)/(x – 2) > 0.
2.4 Definition of instantaneous rate of change at x = a. _____________________ (page 135).
Four Step Process? (page 139)
2.5 Example 7: An object moves along the y axis so that its position at time x (in seconds) is
f(x) = x3 – 6x2 + 9x. Find velocity at x = 2 and x = 5. Find times when velocity is 0.
3.1 Example 1: If $100 is invested at 6% compounded continuously, what amount will be in the
account after 2 years?
3.3 What are the product and quotient rules?___________________ _____________
3.3 Example 2: Find f’(x) for f(x) = (2x – 9)(x2 + 6)
3.3 Example 5: Find f’(x) for f(x) = lnx/(2x + 5)
3.4 What is the Chain Rule __________________ (page 204).
2,
3.4 Example 5: For y = h(x) = e 1 + (lnx) find dy/dx
3.5 (SI20):#35 Find the equation of the tangent line xy – x – 4 = 0; x = 2
3.5 Example 2: Find the equation of the tangent lines to the graph of y – xy2 + x2 + 1 at x = 1.
3.6 Related Rates
p. 220 Example 1: A 26-foot ladder is placed against a wall. If the top of the ladder is sliding
down the wall at 2 ft/second, at what rate is the bottom of the ladder moving away from the wall
when the bottom of the ladder is 10 feet away from the wall?
Example 2: Suppose that two motorboats leave from the same point at the same time. If one
travels north at 15 mph and the other travels east at 20 mph, how fast will the distance between
them be changing after 2 hours?
4.1 Example 4: Find the critical numbers of f, the intervals on which f is increasing, and those on
which f is decreasing, for f(x) = 1/(x - 2)
What is the difference between critical and inflection points? _________________
4.2 Example 3: Find the inflection points of f(x) = ln (x2 – 4x + 5)
4.4 Example 1: Graph f(x) = (x – 1)/(x – 2).
4.5 Example 1: Find absolute max and min f(x) = x3 + 3x2 – 9x – 7 on [-6, 4]
4.6 Example 1: A homeowner has $320 to spend on building a fence around a rectangular
garden. Three sides of the fence will be constructed with wire fencing at a cost of $2 per linear
foot. The fourth side will be constructed with wood fencing at a cost of $6 per linear foot. Find
the dimensions and the area of the largest garden that can be enclosed with $320 worth of
fencing.
5.1 Chapter 5 REVIEW (SI 38)
5.1 ∫(x3 – 3)/x2 dx
5.2 Example 3: ∫(x2 + 2x + 5)5(2x + 2)dx
5.2 ∫4x2√(x3 + 5)dx
5.3 p. 348 Example 4: For a particular person learning to swim, the distance y (in feet) that the
person is able to swim in 1 minute after t hours of practice is given approximately by y = 50(1 –
e-0.04t). What is the rate of improvement after 10 hours of practice?
5.4 Fundamental Theorem of Calculus: ______________________.
5.5 Example 3: 0∫5 x/(x2 + 10) dx =
5.5 Example 4: -4∫1 √(5 – t)dt =
6.1 Example 1 (page 383): Find the area bounded by f(x) = 6x – x2 and y = 0 for 1 ≤ x ≤ 4.
6.1 Example 4 (page 384): Find the area bounded by f(x) = 5 – x2 and g(x) = 2 – 2x.
6.3: Integration by Parts Formula _____________________ *You must know this
Example 1: (page 403) ∫ xex dx
Example 2 (page 404): ∫ x lnx dx