Math 1071Q Exam 1 Review

Math 1071Q
Exam 1 Review
Spring 2013
Disclaimer: This is my attempt to help you study. The problems below are a very minimal set
of problems. You should know the general topics, concepts and problems discussed in class, the
textbook or assigned as homework. As well as be able to do problems related to those found in
the homework problems and quizzes. Problems on the exam may be on topics not covered in this
review sheet and topics here may not appear on the exam. But, this review sheet can give you
some extra practice with most of the main concepts covered in the course.
Functions
You should know what a function is, how to find the domain of a function, how to graph basic
functions and how to read graphs of functions. You should know the different ways to combine
functions, most especially by composing them. You should know how to model some situations
using functions – real-life situations modeled by functions in this class so far include supply and
demand, cost, revenue, profit, interest, compounding interest, break-even quantity, equilibrium
point.
1. Find the domain of each of the following functions:
(a)
√
x+4
√
3
(b) x − 2
p
x2 − 4
(c)
x2 + 1
x2 − 1
x+1
(e)
x−2
x+1
(f) √
x−7
(d)
2. Let f (x) = x3 + 4x2 + 5 and g(x) = 2x2 − 3. Express the following in terms of x:
(a) (f + g)(x)
(b) (f − g)(x)
(c) 3f (x)
(d) (f g)(x)
(e) (f /g)(x)
(f) (f ◦ g)(x)
(g) (g ◦ f )(x)
3. If f (x) =
√
x − 4 and g(x) = x2 , what is (f ◦ g)(−3)? What is the domain of f ◦ g?
4. Find the break-even quantities using the given revenue and cost functions:
R(x) = −3x2 + 20x,
1
C(x) = 2x + 15.
Math 1071Q
Exam 1 Review
Spring 2013
5. Given the demand curve p = −2x + 4000 and the supply curve p = x + 1000, find the
equilibrium point.
6. Suppose that 10,000 units of a certain item are sold per day by the entire industry at a price
of $150 per item and that 8000 can be sold per day by the same industry at a price of $200
per item.
(a) Find the demand equation for p, assuming the demand curve to be a straight line.
(b) Find the revenue function R(x).
(c) Find the value of x for which the revenue is maximized and the maximal revenue.
Exponents and Logarithms
Some new, very important functions introduced in this class are exponential functions and logarithmic functions. You should know what those are, their basic properties, how to manipulate them
and how to solve equations that involve them.
1. Simplify ex ln 5 .
√
x3 y(x2 )5 yz
2. Simplify
.
z −3 x
3. What is the definition of logarithm?
4. Simplify 10log10 5 and log10 (1013 )
5. Write 2 log x + 3 log y − 5 log z as a single logarithm.
6. Solve for x:
(a) 9x = 81.
1
(b) 32x = .
27
2
(c) ex = e5x−6
(d) ln(x2 ) = 9
(e) ln(x2 + 2) = ln(3x).
(f) 7x
2 +2x
= 7−x
(g) 2 log(x + 7) + 3 = 0
7. Solve the following interest problems:
(a) You deposit $2,500 dollars in an account that yields 2% annual interest. How much
money will be in the account after 5 years, if the interest is compounded:
i. monthly?
ii. quarterly?
iii. continuously?
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Math 1071Q
Exam 1 Review
Spring 2013
(b) If you invest $16,000 in a CD and interest is compounded quarterly at 3%, how long will
it take your investment to grow to $20,000?
(c) Suppose $1,000 is invested at an annual rate of 6% compounded continuously. How long
will it take for this amount to double?
Limits
Limits are an important tool. Most especially for this class, they appear in the definition of the
derivative. You should know what a limit is conceptually, as well as how to compute them by
plugging in values, by looking at a graph, and algebraically. You should know the rules of limits,
and how limits relate to continuity of functions.
1. Find the limits. If a limit does not exist, then explain why.
(a) lim (x2 + 3x + 1)
x→1
3x2 + 1
x→−1 x + 2
p
(c) lim
5 − x2
(b) lim
x→−1
x2 − 3x + 2
x→2
x−2
x+1
(e) lim 2
x→−1 x − 1
√
x−2
(f) lim
x→4 x − 4
x2 + 3x + 2
(g) lim
x→−1
x+1
1
(h) lim
x→2 x − 2


−x + 4 if x > 1
2. Let f (x) = 0
if x = 1 . Find limx→1− f (x), limx→1+ f (x), and limx→1 f (x).

 2
x
if x < 1
(
x if x > 0
3. Let f (x) = 1
. Find limx→0− f (x), limx→0+ f (x), and limx→0 f (x).
if x < 0
x
(d) lim
4. If f (x) is given by the picture below, find the following limits:
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Math 1071Q
Exam 1 Review
Spring 2013
(a) limx→1+ f (x)
(b) limx→1− f (x)
(c) limx→1 f (x)
(d) limx→−2+ f (x)
(e) limx→−2− f (x)
(f) limx→−2 f (x)
(g) limx→3+ f (x)
(h) limx→3− f (x)
(i) limx→3 f (x)
(j) limx→4+ f (x)
(k) limx→4− f (x)
(l) limx→4 f (x)
(m) Find all the values a where f (x) is not continuous.
(n) Find all values a where f 0 (a) does not exist.
Rates of Change
You should know what is meant by an average rate of change as well as how to compute an average
rate of change. You should know how to interpret your computations. Same goes for instantaneous
change. You should know how to interpret average and instantaneous change on the graph of a
function, and the difference between them. You should know how to compute the derivative of a
function using limits and how to find the tangent line at a point.
1. Find the average rate of change of the function on the given interval:
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Math 1071Q
Exam 1 Review
Spring 2013
(a) f (x) = ln x, [1, 4].
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(b) f (x) = , [3, 5].
x
(c) f (x) = −x2 + 3x, [0, 3].
2. State the limit definition of the derivative of a function f (x).
3. Use limits to find the instantaneous rate of change and an equation of the tangent line of the
function f (x) = 2x2 − x − 1 at x = 2.
4. Use limits to find the derivative of the following functions:
(a) f (x) = 2x2 − 5x + 1
(b) f (x) = 3 − x2
1
(c) f (x) = .
x
5. Consider the following graph. Which is greater – the average rate of change over the interval
(-1.5,0) or the instantaneous rate of change at x = 2?
Derivatives
You should know the definition of the derivative, what it means, the various ways of interpreting it
as the slope of the tangent line or the instantaneous rate of change. You should be able to compute
it from the definition, or using rules. You should know all of the rules for basic derivatives, products,
quotients, and the chain rule. You should know how the value of the derivative can be related to
the graph of a function and vice versa. You should know how derivatives can be interpreted in
real-world problems. You should know how to use derivatives to find the equation of a tangent line.
1. Find the derivative. You may use whatever rules are appropriate:
(a) f (x) = (x2 + 3x − 2)100 (2x + 5)8
√
3
5
(b) f (x) = 3 x − √ + √
3
x
x2
(c) f (x) =
2x5 − 3
x−5
5
Math 1071Q
Exam 1 Review
Spring 2013
(d) f (x) = 3x−1/3 − 2x1/5 + 4x1/6
p
(e) f (x) = x5 + 3x3 + 10x + 1
x5 + 1
(7x + 3)4
1
(g) f (x) = √
x2 + 2x + 1
(f) f (x) =
2. Find the equation of the tangent line of the given function at the indicated point:
(a) f (x) = x2 − 3x, at the point (2, −2).
1
, at the point with x = 0.
(b) f (x) =
x+2
(c) f (x) = x ln x, at the point with x = 1.
Applications
1. The cost function for a certain product is C(x) =
√
x(x + 1000) + 1000.
(a) Find the marginal cost.
(b) Find C 0 (100) and interpret what it means.
2. Suppose that the price and demand for a certain commodity is given by p(x) = e−x , where x
is the number of items sold in thousands.
(a) Find the marginal revenue.
(b) Where is the marginal revenue positive? Where is it negative? Where is it zero? Interpret what this means.
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