MATH 131:100 Exam 1 Review
10 June, 2014
The first Exam will have three sections: one of Definitions, one for graphical concepts,
and a section of workout problems. For this Review, I will just give a sample of the types
of problems that I could ask, but any of the material we have covered up to Section 2.5 is
fair game.
Definitions You Should Know:
1) A function f is increasing on an interval [a, b] if
f (x1 ) < f (x2 )
whenever
x1 < x2
2) A function f is decreasing on an interval [a, b] if
f (x1 ) > f (x2 )
whenever
x1 < x2
3) A function f is even if
f (−x) = f (x)
for all x
4) A function f is odd if
f (−x) = −f (x)
for all x
5) A function f is one-to-one (or 1-1) if
f (x1 ) 6= f (x2 )
whenever
x1 6= x2
6) A function f is left continuous at a if
lim f (x) = f (a)
x→a−
7) A function f is right continuous at a if
lim f (x) = f (a)
x→a+
8) A function f is continuous at a if
lim f (x) = f (a)
x→a
9) If lim− f (x) = lim+ f (x) = L, then we define the two-sided limit to be
x→a
x→a
lim f (x) = L
x→a
10) lim f (x) = ∞ means that as x grows larger, f (x) grows larger without bound.
x→∞
11) lim f (x) = −∞ means that as x grows larger, f (x) becomes more negative without
x→∞
bound.
Other Topics:
ˆKnow the different types of function classes that we talked about in Section 1.2.
ˆKnow all of the different operations (shifting, stretching, reflecting) from Section 1.3 for
getting new functions from old ones.
ˆKnow how to work with the absolute value function, including its definition.
ˆKnow how to do Exponential growth/decay problems! There will definitely be a problem
like this on the exam.
ˆKnow how to find the graph of an inverse function (i.e. reflecting it about the y = x
line)
ˆKnow how to algebraically solve for the inverse of a one-to-one function.
ˆKnow properties of logarithms and exponential functions, and how they relate to one
another (including cancellation equations)
ˆKnow how to find the average rate of change of a function on an interval. In general, if
you want the average rate of change of a function f (x) on the interval [a, b], the equation is
f (b) − f (a)
b−a
Think about our example finding average velocity from Section 2.1.
ˆKnow Point-Slope Form equation for finding the equation of a line between two given
points.
ˆKnow how to determine left and right-sided limits and two-sided limits from the graph
of a function.
ˆKnow how to determine where a function is increasing, decreasing, and continuous from
its graph.
ˆKnow what the graphs of even and odd functions look like.
ˆKnow the Limit Laws from Section 2.3 and how to apply them to compute limits
algebraically.
ˆKnow how to find the Domain and Range for all of the kinds of functions we have
discussed, and use these to determine where a function is continuous.
ˆKnow how to take limits of functions to ∞ and −∞.
ˆKnow how to determine if the one-sided limit of a function is +∞ or −∞ at a vertical
asymptote.
ˆKnow how to find Horizontal and Vertical asymptotes of functions.
Example Problems:
Problem 1) The following table shows the population of the world every decade from
1900 to 2000.
Year
Population (millions)
1900
1650
1910
1750
1920
1860
1930
2070
1940
2300
1950
2560
1960
3040
1970
3710
1980
4450
a) What was the average rate of change of the population from 1940 to 1980?
b) Suppose you want to model the data given with a linear model. Determine the linear
model to approximate Population, P , as a function of time, t, in years by using the data
values for 1950 and 2000.
c) Use your model from part (b) to estimate the population of the world in the year
3000.
Problem 2) On the same graph, sketch the functions y = ln x and y = ln x + 1.
Problem 3) On the same graph, sketch the functions y = x3 and y = (x + 3)3 − 2.
1990
5280
2000
6080
Problem 4) Consider the function f (x) = ln(x2 − 1) for x > 1. That is D(f ) = (1, ∞).
You are given that the function is one-to-one on its domain, so it has an inverse. Find f −1 (x)
and its Domain and Range.
Problem 5) Solve the following equation for x.
2 + log2 (x + 5) = log2 (5x − 6)
Problem 6) What are the Domain and Range for f (x) = ln(x − 5) + 6?
Problem 7) Evaluate the following limit, if it exists.
2x2 + 3x + 1
x→−1 x2 − 2x − 3
lim
Problem 8) Evaluate the following limit, if it exists.
lim
2−x
− 1)2
x→1 (x
Problem 9) Evaluate the following limit, if it exists.
2
lim e−x
x→∞
Problem 10) Evaluate the following limit, if it exists.
1
1
lim
−
x→1 2(x − 4)
2(x − 2)