Download review questions

Math 116.05 - Final Review Answers
The following will be included with the exam.
To Graph:
Draw the graph of f and:
Functional Change to f (x
y = f (x) + k, k > 0
Raise the graph of f by k units.
Add k to f (x).
y = f (x) − k, k > 0
Lower the graph of f by k units.
Subtract k from f (x).
y = f (x + h), h > 0
Shift the graph of f to the left h units.
Replace x by x + h.
y = f (x − h), h > 0
Shift the graph of f to the right h units.
Replace x by x − h.
Multiply each y-coordinate of y = f (x) by a.
Multiply f (x) by a.
Vertical shifts
Horizontal shifts
Compressing or stretching
y = af (x), a > 0
Stretch the graph of f vertically if a > 1.
Compress the graph of f vertically if 0 < a < 1.
y = f (ax), a > 0
Multiply each x-coordinate of y = f (x) by a1 .
Replace x by ax.
Stretch the graph of f horizontally if 0 < a < 1.
Compress the graph of f horizontally if a > 1.
Reflection about the x-axis
y = −f (x)
Reflect the graph of f about the x-axis.
Multiply f (x) by −1.
Reflect the graph of f about the y-axis.
Replace x by −x.
Reflection about the y-axis
y = f (−x)
1
2
−b ±
√
b2 − 4ac
2a
P (1 + r)t
r n ∆t
P 1+
n
P ert
R·
(1 + r)n − 1
r
1 − (1 + r)−n
r
n!
n Pr =
(n − r)!
R·
n Cr
=
n!
(n − r)!r!
Math 116.05
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(1) Chapter 2 - Functions
(a) Section 2.1 - Functions
(i) What three things make up a function?
A domain (a collection of inputs), a range (a collection of outputs), and a
rule that assigns an output for each input.
(ii) Draw a diagram of a function that sends 2 to A, 4 to X, and 16 to H. What
is the domain and range of the function?
Domain :
Range :
2
4
16
A
X
H
The domain is {2, 4, 16}, and the range is {A, X, H}.
(iii) If x is a person, let f (x) be the parent of x. Is f a function? Why or why
not?
Each person has two (biological) parents, so each input to f will have two
outputs. But this isn’t allowed for functions. So f is not a function.
(iv) If x is a person, let g(x) the the child of x. Is f a function? Why or why
not?
Not every person has a child, so not every input to g has an output. But
this isn’t allowed for functions. So g is not a function.
(v) What is the domain of ln(2x − 3)?
For this formula to make sense, we must have 2x − 3 > 0, so the domain is
x > 32 .
(vi) What is the domain of
The domain is x ≤
p
−(3x + 16)?
16
3 .
(vii) True or False: for any x and y and any function f , we have f (x + y) =
f (x) + f (y). Explain why your answer is correct.
False! Very few functions distribute like this.
(viii) Given the equation xy 2 = x, is x a function of y? Is y a function of x?
2
No and no. The collection of points determined by this equation is the
union of the lines x = 0, y = 1 and y = −1.
(b) Section 2.2 - Types of functions
(i) Define the following kinds of functions: constant, linear, polynomial, rational.
(A) constant: f (x) = c for some number c.
(B) linear: f (x) = mx + b for some m and b.
(C) polynomial: f (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 for some
constants ai .
(D) rational: f (x) = p(x)/q(x) for some polynomials p(x) and q(x).
(ii) Sketch
of the following functions: x, x2 , x3 , xn for a positive integer
√
√ graphs
n, x, 3 x, 1/x, 1/x2 , 1/x3 , 1/xn for a positive integer n, and |x|.
Use Worlfram Alpha or Geogebra to get graphs for these functions.
(iii) True or false: all polynomial functions are rational.
True.
(iv) True or false: all rational functions are polynomials.
False.
(v) Write a function that is not a polynomial and not rational.
For example, f (x) =
√
x is not a polynomial and not rational.
(c) Section 2.3 - Combining functions
(i) Define the sum, difference, product, and quotient of two functions f (x) and
g(x).
(f + g)(x) = f (x) + g(x), (f − g)(x) = f (x) − g(x), (f g)(x) = f (x)g(x),
(f /g)(x) = f (x)/g(x).
(ii) If f (x) = 2x2 − 6x + 1 and g(x) = 2x − 1, write (and simplify) formulae for
the functions (f · g)(x) and (f − g)(x).
(f · g)(x) = 4x3 − 14x2 + 8x − 1 and (f − g)(x) = 2x2 − 8x + 2.
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(iii) How is the domain of (f − g)(x) related to the domain of f (x) and g(x).
The domain of f − g is the intersection of the domain of f and the domain
of g.
(iv) If q(t) is the quantity of a product sold by a company in month t, and p(t)
is the price of the product in month t, write a formula for the renevue r(t)
the company makes in month t.
r(t) = p(t) · q(t)
(d) Section 2.4 - Composition of functions and inverse functions.
(i) What is the definition of the composition of two functions f (x) and g(x)?
What is the notation for “composition”? (I.e. what symbol do we put
between f and g to indicate that they should be composed?)
(f ◦ g)(x) = f (g(x)).
(ii) If f (q) gives the revenue in dollars for selling q items (in units), and g(m)
gives the quantity (in units) produced by m employees (in individuals),
which of the compositions f ◦ g and g ◦ f make sense? For those that make
sense, what are the units for the input and output of the composition?
Only (f ◦ g)(m) makes sense. The input m is a number of employees and
the output is revenue in dollars.
(iii) If f (x) = 2x2 − 6x + 1 and g(x) = 2x − 1, write a formula for (f ◦ g)(x).
(f ◦ g)(x) = 8x2 − 20x + 7
(iv) What is the definition of the inverse function of a function f (x)? I.e. g(x)
is the inverse of f (x) if...?
g(x) is the inverse of f (x) if f (g(x)) = x for all x in the domain of g, and
g(f (x)) = x for all x in the domain of f .
(v) Find the inverse of f (x) =
√
4 − 2x.
f −1 (x) = − 12 x2 + 2
(vi) What does it mean for a function to be one-to-one?
A function is one-to-one if every output has a single input that produces
that output. In symbols, f (x) is one-to-one if f (a) = f (b) implies a = b.
4
(vii) What is the horizontal line test, and what does it tell you about functions?
A graph passes the horizontal line test if every horizontal line intersects the
graph at most once. If the graph of a function passes the horizontal line
test, then the function is one-to-one.
(viii) What is the relationship between a function being one-to-one and having
an inverse?
A function has an inverse if and only if it is one-to-one.
(ix) What is the inverse of the function g(x) = −3x2 + 5?
This function does not have an inverse.
(x) What is the inverse of the function h(x) = 2x2 + 6 for x ≤ 0?
q
h−1 (x) = − 12 x − 3
(xi) How is the graph of f −1 (x) related to the graph of f (x)? Use this to sketch
a graph of the inverse of f (x) = x2 + 1 for x > 0.
The graph of f −1 (x) is the same as the graph of f (x), but reflected across
the line y = x.
(e) Section 2.5 - Graphs of Functions
(i) What is the definition of the graph of a function? I.e. the graph of a
function f (x) consists of points in the plane such that...?
Math 116.05
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The graph of a function f is the collection of points of the form (x, f (x)).
Another way to say this is that the graph is the collection of points whose
coordinate satisfy the equation y = f (x).
(ii) Sketch a graph of the function f (x) =
√
x2 + 4
Note that this graph is not a parabola, although it might seem like one,
depending on how many points you plot.
(f) Section 2.6 - Symmetry and Reflections
(i) How do you tell if a function f (x) is even (i.e. its graph is symmetric about
the y-axis) or odd (i.e. its graph is symmetric about the origin)?
A function f is even if it satisfies the equation f (x) = f (−x) for all x in the
domain of f . A function f is odd if it satisfies the equation f (−x) = −f (x)
for all x in the domain of f .
(ii) Are the following even, odd, or neither?
√
3
x,
x
,
x+3x3
e−x .
Odd and not even, even and not odd, neither.
(iii) Are the graphs of the following equations symmetric about the y-axis, the
x-axis, the origin, or none of the above?
x=y
x3 = y
x2 + y 3 = 1
x/y 2 = 6.
(A) x = y: symmetric across both axes and through the origin.
(B) x3 = y: symmetric through the origin.
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(C) x2 + y 3 = 1: symmetric across the y axis.
(D) x/y 2 = 6: symmetric across the x axis.
(g) Section 2.7 - Shifting, Stretching,
√ and Reflecting Graphs
(i) Starting with a graph of x, sketch a graph of each
√ of the following functions using a graphical operation. (Start over with x for each one.) Check
your √
answer using a graphing calculator, Wolfram Alpha, or Geogebra.
(A) x + 4
(B)
√
x−9
(C) 1 +
(D)
√
√
x
x−4
√
(E) 2 x
(F)
√
2x
Check your answers with Wolfram Alpha or Geogebra.
(ii) If we start with the function f (x) = e2x and we reflect its graph about the
y axis, what function do we get a graph of? What if we reflect the graph
of f (x) through the origin?
If we reflect across the y axis, we get a graph of e−2x . If we reflect through
the origin, we get a graph of −e−2x .
(iii) Starting with a graph of 1/x, carry out the following operations, one after
the other, in the given order. (Do not start over with a graph of 1/x at
each step.) At each step, write a formula for the graph in that step.
(A) Shift up 2.
(B) Stretch vertically by a factor of 3
(C) Reflect across the y-axis.
(D) Shift left 4.
Shifting up 2 gives a graph of
1
x
+ 2:
Math 116.05
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Stretching verically by a factor of 3 gives a graph of 3
Reflecting across the y axis gives a graph of
3
−x
+ 6.
1
x
+2 =
3
x
+ 6.
8
Shifting left 4 gives a graph of
3
−(x+4)
+6=
3
−x−4
+ 6.
(2) Chapter 3 - Linear and quadratic functions, applications
(a) Section 3.1 - Linear Functions
(i) What is a linear function?
A function f (x) is linear if it is of the form f (x) = mx + b for constants m
and b.
(ii) Which of the following functions are linear?
f (x) = − 12 x + 4,
g(x) =
2x − 1
,
6
h(x) = 1/x.
Math 116.05
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Only the first two are linear.
(iii) Is the data in the table below given by a linear function y = f (x)? If so,
find the function. If not, explain why not.
x 2 4 5
9
y 8 2 −1 −13
Yes. This is given by the function f (x) = −3x + 14.
(iv) What is the domain and range of a linear fuction?
The domain of a linear function is all real numbers. The range of a linear
function is all reals if m 6= 0. If m = 0, the range is just the number b.
(v) Are linear functions one-to-one? Invertible?
A linear function mx + b is one-to-one (and invertible) if m 6= 0.
(vi) If there are q tons of a good available in a market, the resulting price is
p(q) = 100 − 12 q. What q values make sense? What q value produces a
price of $0.50?
The only q values that make sense are 0 ≤ q ≤ 200. For a price of $0.50,
we need q = 199.
(b) Section 3.2 Applications of Linear functions
Pick any problems from section 3.2 for practice using linear functions.
(c) Section 3.3 - Quadratic Functions
(i) What is the domain of a quadratic function?
All reals.
(ii) Are quadratic functions one-to-one? Invertible?
No and no.
(iii) For the quadratic f (x) = − 12 x2 − 8x + 2, complete the square and use your
result to sketch a graph.
f (x) = − 12 (x+8)2 +34. So, starting with a graph of x2 , we do the following
operations:
(A) shift left 8.
(B) Flip across the x axis.
10
(C) Compress vertically by 2.
(D) Shift down 30.
(iv) Where is the vertex of x2 + 4x + 5?
At the point (−2, 1).
(v) What is the range of g(x) = −x2 + 2x − 2?
(−∞, −1]. (Note the right square bracket.)
(vi) If a company manufactures q gallons of paint, then they can sell it for
1
q dollars per gallon. How much paint should the company
p(q) = 500 − 30
manufacture to maximize its revenue?
7500 gallons.
(d) Section 3.4 - Systems of Linear Equations
(i) What is a linear system?
A system of equation where each equation is linear, i.e. of the form
ax + by + cz + ... = d.
(Here, a, b, c, and d are all constants, and x, y, and z are variables.)
(ii) How many solutions can a system of linear equations have?
Math 116.05
11
One, zero, or infinitely many (in which case we might have a one, two, or
more parameter family of solutions.)
(iii) Solve the system


x − y + z
2x + 6y − 3z


x+y+z
= 10
= 18
= 3.
7
26
( 117
10 , − 2 , − 5 )
(iv) How can you tell if your system has no solutions? How can you tell if it
has infinitely many solutions?
If there are no solutions, you will get an equation that can never be satisfied,
something like “1 = 0”. If there are infinitely many solutions, you will get
the equation “0 = 0”.
(v) Solve the system


2x + y − z
x−y−z


x + 2y
=4
=1
= 3.
The set of solutions is the one parameter family ( 23 z + 53 , − 13 z + 23 , z).
(vi) If you want to mix a 5% acid solution and a 20% acid solution to get 200
liters of 14% acid solution, how much of each solution should you use?
80 liters of the 5% solution and 120 liters of the 20% solution.
(e) Section 3.5 - Nonlinear systems
(i) How many solutions can a nonlinear system have?
Any number of solutions.
(ii) When you have solved a nonlinear system, how can you check if your results
are correct?
You can check that your solutions satisfy the original equations.
(iii) Solve the system
(
2x + 3y
y
=3
= x2 − 4x.
12
There are two solutions:
√
√ 5
1
1
2
34,
−
and
−
+
3
3
9
9 34
5
3
+
1
3
√
34, − 19 −
2
9
√
34 .
(iv) Solve the system
(
1
x
y
= 2y
√
= x + 3.
As written, to solve this system you need to solve a cubic equation. This
is possible, but this is something we didn’t talk about, and I won’t ask you
to do so on the exam. Let’s solve the following system instead:
(
1
= 2y
x
y = x + 3.
Multiplying the first equation by 1/2, we get 1/(2x) = y. Now we have
both equations solved for y, so we can put them together:
1
=x+3
2x
1 = 2x2 + 6x
0 = 2x2 + 6x − 1
√
−6 ± 44
x=
4√
−3 ± 11
.
=
2
(f) Section 3.6 - Applications of systems of equations.
(i) If the supply and demand equations in a market are
(
1 2
10 q − q + 15 = p
p
= − 15 q + 100
(with q in units and and p in dollars), what price and quantity does the
market settle on? (I.e. what is the price and quantity at any market equilibria?)
There is only one equilibrium (with positive q and p), q = 33.4279, p =
93.31.
(ii) Continuing the previous problem, how does the equilibrium change is the
government ads a $1 per unit subsidy for this product?
The new market equilibrium is q = 33.2575 and p = 93.35.
(3) Chapter 4 - Exponential functions and logarithms
(a) Section 4.1 - Exponential Functions
Math 116.05
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(i) What is an exponential function?
A function of the form f (x) = Abx for some constants A and b, with b > 0.
(ii) What are three examples of situations described by exponential functions?
Compound interest in a bank account, population growth, and radioactive
decay.
(iii) What is the domain and range of and exponential function bx , if b is a
constant satisfying b > 0 and b 6= 1?
The domain is all reals, and the range is (0, ∞).
(iv) If a 5kg radioactive sample has a halflife of one day, write a function m(t)
that gives the mass of the original sample remaining after t days.
m(t) = 5( 21 )t .
(v) What is the definition of the number e?
e is the number the expression
1 n
1+
n
tends towards as n becomes very large.
(b) Section 4.2 - Logarithms
(i) When is an exponential function of the form bx for a constant base b > 0
one-to-one? When it is one-to-one, what is its inverse?
It is one-to-one when b 6= 1, and the inverse is logb (x).
(ii) Rearrange the equation xy = z using a logarithm.
logx (z) = y
(iii) Sketch a graph of the functions log3 (x) and log1/3 (x).
14
(iv) What is the domain and range of loga (x)? How does this depend on the
base a?
The domain is (0, ∞) and the range is all reals. This does not depend on
a.
(v) What is meant by “ln(x)” and “log(x)”?
“natural log”, i.e. log base e, and “common log”, which is log base 10.
(vi) What bases does your calculator have a log button for?
Most calculators only have buttons for natural log and common log.
(vii) Find the following logarithms without using a calculator:
log3 (9)
log1 0(1000)
1
log4 ( 16
ln(eπ )
2, 3, −2, π
(viii) If a population grows according to the formula P (t) = 1000e0.05t (where t
is in years and P is in individuals), how long goes it take for the population
to reach 100000?
It takes 92.1034 . . . years.
(c) Section 4.3 - Properties of Logarithms
(i) List as many properties of logarithms and formulae involving logarithms as
you can. Compare your list to the one in the book. Did you forget any?
Check in the book!
Math 116.05
15
(ii) Rewrite the following, writing exponents as factors.
2
u
3 1
log3 (3x y 5)
ln
v4
1 + 3 log3 (x) + 15 log3 (15) and 2 ln(u) − 4 ln(v).
(iii) Using a calculator, find log4 (10). How can you check your answer?
1.660964.... I can check this by raising 4 to this power and verifying that I
get 10.
(iv) Rewrite the following using a single logarithm:
log4 (x2 y 4 ) + log2 (x3 y)
log2 (x7 y 9 ). If your answer has a different base for the logarithm, it might
look a bit different.
(d) Section 4.4 - Logarithmic and Exponential Equations
(i) Solve the following equations:
2
ex e−3x e2 = 1
ln(3x − 2) + ln(x + 1) = 0
ln(eln(x) ) = e
x = 1, 2
√
−1 ± 37
x=
6
e
x=e
(4) Chapter 5 - Finance
Annuities due (from section 5.4) and all of section 5.5 will not be on the final exam.
(a) If I want to deposit a lump-sum in an account earning 6% annual interest (compounded annually) so that I will have $1500 in four years, how much must I
deposit? How does this change if interest in compounded continuously?
Annual compounding: $1188.14. Continuous compounding: $1179.94.
(b) If I have $2000 now and in ten years I have $5500, what effective annual interest
rate did I get? If interest was actually compounded daily, what nominal annual
interest rate did I get?
Effective rate: 10.645%. Nominal rate: 10.1174%.
(c) I want to prepay my car insurance for five years. If my car insurance payment is
$250 semi-annually (at the end of each 6 month period), and the interest rate is
4% compounded semiannually, how much must I pay now?
$2249.65
16
(d) I want to sponsor an annual footrace that will take place once per year for 10
years. I estimate that the race will cost $10000 to organize each year it is run.
If I can get 5% effective annual interest, and I can save $5000 per year, how long
will it take me to save enough to fund all 10 annual races?
Rounding up, 12 years.
(e) What is the effective interest rate of a) 7% interest compounded annually, b) 3%
interest compounded every hour, and c) 8% compounded continuously?
a) 7%, b) 3.045448%, c) 8.3287%
(f) How long does it take to double your money in an account bearing 2.5% interest
compounded annually?
27.071 years.
(g) A lottery winner is offered $1000000 now, or a $2000 every year forever. If the
winner can get 3% annual effective interest, which option should they take?
Take the cash.
(5) Chapter 8 - Counting and Probability
(a) What is the difference between n Cr and n Pr ?
n Pr
distinguishes the order of the chosen objects, n Cr does not.
(b) How many ways are there to draw a straight (5 cards of consecutive rank, ignoring
suit) in a 5-card poker hand?
9216.
(c) What is the probability of drawing a straight?
0.0035460
(d) Arnold Schoenberg created the twelve-tone technique for composing music, wherein
each of the 12 notes of the chromatic scale was used once before any other notes
were repeated. How many 12 note twelve-tone melodies could Schoenberg have
written before he had to stop?
12!
(e) A powerball drawing has 5 numbers from 1 to 59 (order is not important for this
part), and a powerball number, drawn from a bin of red balls numbered 1 to 35.
Math 116.05
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What is the sample space for this lottery drawing, and what is its size? If you
buy one ticket, what is your probability of winning?
A single element of the sample space is a combination of 5 numbers from 1 to
59 together with a single number from 1 to 35. The sample space has 175223510
elements. The probability of winning is 5.7 ∗ 10−9 .
(f) What is the probability of rolling a total of 7 on two 6-sided dice? What is the
probability of rolling a total of 11?
1
Rolling a 7: 61 . Rolling an 11: 18
.
(g) Craps is played by rolling two 6-sided dice. Rolling a 7 or 11 as your first roll
wins. What is the probability of winning on your first roll?
2
9
(h) What is the probability of drawing a heart or a 3 from a 52-card deck?
4
13