Final Exam Study Guide and Practice Problems

Final Exam Coverage: Sections 10.1 - 10.2, 10.4 - 10.5, 11.2 - 11.5, 12.1 - 12.6, 13.1 - 13.2,13.4 - 13.5, 14.1, and 15.1 - 15.2.

Sections NOT on the exam: Sections 10.3 (Continuity), 10.6 (Differentials), 10.7 (MarginalAnalysis in Business and Economics), 11.1 (The constant e and Continuous Compound Interest),11.6 (Related Rates), 11.7 (Elasticity of Demand), and 14.2 (Applications in Business and Eco-nomics)

List of Topics for the Final Exam

Section 10.1: Introduction to Limits

� Know what a limit means (you don’t need to know the exact definition but you should havea general idea of what is meant by taking a limit of a function)

� One-sided limits

� When does a limit limx→c

f(x) fail to exist?

� Know how to use a graph of a function to determine various limits or show that a limit doesnot exist

� Know properties of limits and how to use them to find limits

� To find a limit as x → c of a rational expression p(x)/q(x), when are you allowed to justplug in c into the expression to get the limit?

� Know what is meant by a 00

indeterminate form and what this means about a limit

� Know how to use factoring to find a limit

� Know when a limit of a rational expression p(x)/q(x) does not exist (or, as discussed in latersections, is +∞ or −∞).

Section 10.2: Infinite Limits and Limits at Infinity

� Infinite limits (such as limx→c

f(x) =∞ or limx→c

f(x) = −∞)

� Know how to identify all vertical asymptotes of a function

� Limits at infinity, such as limx→∞

f(x) = L or limx→−∞

f(x) = L. We could also replace L in these

expressions with ∞ or −∞.

� Know how to find horizontal asymptotes of a function

� Know how to take limits at infinity of polynomial functions and rational functions.

Section 10.4: The Derivative

� You don’t need to know the exact definition of the derivative as a limit, but you do needto know the different interpretations of the derivative (e.g. the derivative f ′(x) can beinterpreted as the slope of the tangent line to f(x) at the point (x, f(x)).)

� Know how to find the equation of the tangent line to a function at a point.

Section 10.5: Basic Differentiation Properties

� power rule

� constant multiple property

� sum and difference properties

Section 11.2: Derivatives of Exponential and Logarithmic Functions

� Know derivative of ex and ln(x)

1

Section 11.3: Derivatives of Products and Quotients

� Product rule

� Quotient rule

Section 11.4: Chain Rule

� Know how and when to use the chain rule

Section 11.5: Implicit Differentiation

� Know how to use implicit differentiation to find dydx

in an equation where y is implicitlydefined

� Know how to evaluate your answer for dydx

at a given point to find the slope of a curve atthat point

� Know how to find the tangent line to a curve at a point

Section 12.1: First Derivative and Graphs

� Critical numbers - know how to find them and how they are useful

� Know how to make a sign chart for f ′(x)

� Know how to use the sign chart to find local maxes/mins

� Know how to find the intervals on which a function is increasing/decreasing

Section 12.2: Second Derivative and Graphs

� Know how to make a sign chart for f ′′(x)

� Know how to use the sign chart to find inflection points and to find on what intervals afunction is concave up/down

� Know how to use information about concavity and increasing/decreasing to help sketch agraph

Section 12.3: L’Hopital’s Rule

� Know what is meant by0

0and

∞∞

indeterminate forms

� Know how to determine if L’Hopital’s Rule can be applied to find a limit of a function

� Know how to use L’Hopital’s rule to find the limit of a function (assuming L’Hopital’s rulecan be used)

Section 12.4: Curve-Sketching Techniques

� Know how to apply the graphing procedure taught in this section which includes

– Looking at f(x) and finding its domain, intercepts, asymptotes (horizontal and vertical)

– Analyzing the first derivative f ′(x) (finding where f(x) is increasing/decreasing, criticalpoints, the local mins/maxes)

– Analyzing the second derivative f ′′(x) (finding the inflection points, where f(x) isconcave up/down)

– Sketching the graph using all of the info obtained

� Be able to draw a sketch of a graph that satisfies certain listed info, which may includeinfo about asymptotes, limits, intervals on which the function is increasing/decreasing, andintervals on which the function is concave up/down.

Section 12.5: Absolute Maxima and Minima

� Know what is meant by an absolute maximum, an absolute minimum, and absolute extrema

� Know how to find (and show that you’ve found) an absolute extrema, either on a closedinterval or an open one

2

Section 12.6: Optimization

� Given an optimization problem, know how to

– Draw a picture (if necessary) and introduce your variables

– Find your constraint equation (an equation that tells how the variables you introducedpreviously are related)

– Write down your objective function (the function you are trying to maximize or mini-mize) as a function of one variable only

– Maximize (or minimize) the objective function, and show that you’ve done so by in-cluding a sign chart. Finish answering the question

Section 13.1: Antiderivatives and Indefinite Integrals

� Antiderivatives (know what they are and how to find them)

� Know how to use integral notation∫f(x) dx to represent the family of antiderivatives of

f(x)

� Properties of indefinite integrals

Section 13.2: Integration by Substitution

� Know how to use integration by substitution (also known as u-substitution) to find anintegral

Section 13.4: The Definite Integral

� You don’t need to know how to compute Riemann sums

� You do need to know what the definite integral represents geometrically in terms of areas.

� Given areas of certain shaded areas of a graph, know how to use these to find a definiteintegral

� Properties of definite integrals

Section 13.5: The Fundamental Theorem of Calculus

� Know what the Fundamental Theorem of Calculus says and know how to apply it to find adefinite integral

Section 14.1: Areas Between Curves

� If f(x) ≥ g(x) over an interval [a, b], know how to find the area bounded by these twofunctions over [a, b].

Section 15.1: Functions of Several Variables

� Given a function f(x, y) of two variables, know how to evaluate the function at certain values

Section 15.2: Partial Derivatives:

� Know how to find first-order partial derivatives and second-order partial derivatives of afunction f(x, y) of two variables

3

Practice Problems

Note: These problems are just some of the types of problems that might appear on the exam.However, to fully prepare for the exam, in addition to making sure you can do all of these prob-lems, you should review your homework, quizzes, and class notes as well. Note problems differentthan the ones listed here may appear on your exam.

Section 10.1: Introduction to Limits

1. The graph of f is shown below. Use the graph of f to evaluate the indicated limits andfunction values. If a limit or function value does not exist, write “DNE.”

(a) limx→−2+

f(x)

(b) limx→−2−

f(x)

(c) limx→−2

f(x)

(d) f(−2)

(e) limx→0

f(x)

(f) f(2)

(g) limx→2−

f(x)

(h) limx→2+

f(x)

(i) limx→2

f(x)

2. Suppose limx→1

f(x) = −3 and limx→1

g(x) = 2. Find

(a) limx→1

3f(x)

(b) limx→1

(g(x)− f(x)

)(c) lim

x→1

3− f(x)

4g(x)− 2

3. Find each limit

(a) limx→3

(x2 − 3

)(b) lim

x→15

(c) limx→2

x

(d) limx→−1

x2 + 1

x

4. For each limit, is the limit a 00

indeterminant form? Find the limit or explain why it doesnot exist.

(a) limx→7

(x− 7)2

x2 − 4x− 21

(b) limx→2

x− 2

x + 2

4

(c) limx→4

3x + 12

x2 − 4

(d) limx→3

x2 − 9

x− 3

5. Circle true or false to the following statements.

(a) If limx→1

f(x) = 0 and limx→1

g(x) = 0, then limx→1

f(x)

g(x)does not exist.

true false

(b) If limx→1

f(x) = 2 and limx→1

g(x) = 2, then limx→2

f(x)

g(x)= 1

true false

(c) In order for limx→1

f(x) to exist and be equal to 2, it must be the case that limx→1−

f(x) = 2

and limx→1+

f(x) = 2.

true false

(d) If f is a function such that f(2) exists, then limx→2

f(x) exists.

true false

(e) If f is a polynomial, then limx→c

f(x) = f(c) for every real number c.

true false

Section 10.2: Infinite Limits and Limits at Infinity

1. Suppose f(x) =1

x− 5. Find each limit. Write ∞, −∞, or DNE where appropriate.

(a) limx→5−

f(x)

(b) limx→5+

f(x)

(c) limx→5

f(x)

2. Suppose f(x) =3− x

x + 2. Find each limit. Write ∞, −∞, or DNE where appropriate.

(a) limx→−2−

f(x)

(b) limx→−2+

f(x)

(c) limx→−2

f(x)

3. Identify all vertical and horizontal asymptotes of the following functions.

(a) f(x) =1

x + 3

(b) f(x) =x2 − 2x− 3

x2 − 9

4. Find the following limits. Write ∞ or −∞ where appropriate.

(a) limx→∞

(x2 − 2x− 1

)(b) lim

x→∞

(− x3 − x

)(c) lim

x→−∞

(x3 + 2x + x

)

5

5. Circle true or false to the following statements.

(a) A polynomial function of degree greater than or equal to 1 has neither horizontal norvertical asymptotes.

true false

(b) A rational function always has at least one vertical asymptote.

true false

(c) A rational function has at most one horizontal asymptote.

true false

6. A theorem states that for n ≥ 1 and an 6= 0

limx→−∞

(anx

n + an−1xn−1 + · · ·+ a0

)= ±∞

What conditions must n and an satisfy for the limit to be +∞?

Section 10.5: Basic Differentiation Properties

1. If C is any constant, what isd

dx(C)?

2. Write down the power rule.

3. If f ′(x) = 2x and g′(x) = x2, what isd

dx

(2f(x)− 1

4g(x)

)?

4. Find the derivative of the following functions. Do not simplify.

(a) f(x) = 7

(b) f(x) = 2x + 3

(c) f(x) = 2x2 − x + 1

(d) y =√x + x3/4

(e) y =1

5x3− 2x− 3

x−2/3

Section 11.2: Derivatives of Exponential and Logarithmic Functions

1. Findd

dx

(5ex − ln(x) + 2

).

2. Use log properties to find derivatives of the following functions. (Note: If you use the logproperties you won’t need to use the chain rule.)

(a) f(x) = x4 − ln(x5)

(b) y = ln(xex)

(c) y = ln( x

5√x

)Section 11.3: Derivatives of Products and Quotients

1. Fill in the blanks:

(a) The product rule says that given two differentiable functions F (x) and S(x), the deriva-tive of their product is

d

dx

(F (x)S(x)

)=

(b) The quotient rule says that given two differentiable functions T (x) and B(x), the deriva-tive of their quotient is

d

dx

(T (x)

B(x)

)=

6

2. Findd

dx

(x2ex

). Do not simplify.

3. If f(x) =x2

4x− 3, find f ′(x). Do not simplify.

4. Find f ′(z) if f(z) =z ln(z)

z3. Do not simplify.

5. Findd

dt

(et√t− 2

3 ln(t) + t3/2

). Do not simplify.

6. Find f ′(x) if f(x) =ex − ln(x)

1− ex. Do not simplify.

Section 11.4: The Chain Rule

1. Find the indicated derivatives. You don’t have to simplify.

(a) f ′(x) if f(x) = (3x2 − 2x)10

(b)dy

dxif y = ln(x2 + 1)

(c) f ′(w) if f(w) =1

3√

(2w − ew)4

(d)d

dt

((1− t)2et

3−t)

Section 11.5: Implicit Differentiation

1. Use implicit differentiation to finddy

dxif 2y − xy2 + xy = 1.

2. Suppose e2xy − x5 + x ln y = 0. Finddy

dx.

3. A curve is described by the equation y + x2 − 2 + xy = 1.

(a) Use implicit differentiation to finddy

dx.

(b) Finddy

dx

∣∣∣(1,1)

. That is, finddy

dxwhen x = 1 and y = 1.

(c) Find the equation of the tangent line to the curve at (1, 1).

Section 12.1: First Derivative and Graphs

1. For each function, find the intervals on which f(x) is increasing and the intervals on whichf(x) is decreasing. Find any local maxes or mins.

(a) f(x) = 2x2 − 8x + 9

(b) f(x) = −x4 + 50x2

(c) f(x) = x3 + 3x2 + 3x

2. Use the given info to sketch the graph of f . Assume its domain is (−∞,∞).

� f(−2) = 4, f(0) = 0, f(2) = −4

� f ′(−2) = 0, f ′(0) = 0, f ′(2) = 0

� f ′(x) > 0 on (−∞,−2) and (2,∞)

� f ′(x) < 0 on (−2, 0) and (0, 2)

Section 12:2: Second Derivative and Graphs

1. Find the inflection points of f(x) = ln(x2 − 4x + 5).

2. Suppose f(x) = x1/3. Find the inflection point(s) of f(x). Also, find the intervals on whichf(x) is concave up and the intervals on which f(x) is concave down.

3. Suppose f(x) = x4 − 2x3. Find

7

(a) The domain of f(x).

(b) The intercepts of f(x).

(c) Make a sign chart for f ′(x). Find its critcal numbers. Find any local extrema. List theintervals on which f is increasing and the intervals on which f is decreasing.

(d) Make a sign chart for f ′′(x). Find any inflection point(s) and list the intervals on whichf(x) is concave up or concave down.

(e) Sketch the graph of f .

Section 12.3: L’Hopital’s Rule

1. For each limit, determine if the limit is a0

0indeterminate form, a

∞∞

indeterminate form,

or neither. Can L’Hopital’s Rule be applied to find the limit? If not, explain why not. If itcan be applied, use L’Hopital’s Rule to find the limit.

(a) limx→0

ln(1 + x2)

x4

(b) limx→1

lnx

x

(c) limx→−∞

ln(1 + 2ex)

ex

(d) limx→∞

e−x

lnx

(e) limx→0

3x + 1− e3x

x2

2. For each limit listed, first (i) find the limit using previous methods, and then (ii) useL’Hopital’s Rule to find the limit.

(a) limx→∞

2x2 − 1

x2 + 1

(b) limx→∞

3x− 1

x2 − 4

(c) limx→−3

x2 − 9

x + 3

Section 12.4: Curve Sketching Techniques

1. Use the graphing strategy taught in this section to analyze and graph the following functions.

(a) g(x) =4x + 3

x2

(b) f(x) = xe−0.5x

(c) f(x) = x lnx

2. Sketch a function which satisfies the following properties.

� f(−3) = −1, f(0) = 0, f(3) = 1

� f ′(x) < 0 on (−∞,−2) and (2,∞)

� f ′(x) > 0 on (−2, 2)

� f ′′(x) < 0 on (∞,−2) and (−2, 0)

� f ′′(x) > 0 on (0, 2) and (2,∞)

� vertical asymptotes: x = −2, x = 2

� horizontal asymptotes: y = 0

Section 12.5: Absolute Maxima and Minima

1. Find the absolute maximum and absolute minimum of f(x) = x3 − 12x on each of thefollowing intervals:

(a) [−3, 3]

(b) [−3, 1]

8

2. Find the absolute extrema of each function on (0,∞).

(a) f(x) = 12− x− 5

x(b) f(x) = f ln(x)− x

Section 12.6: Optimization

1. Find the greatest possible product of two numbers given that the sum of the two numbersequals 28.

2. Find two numbers whose difference is 80 and whose product is a minimum.

3. Find the area of the largest rectangle that can be made with a perimeter of 56 ft.

4. Find the largest possible perimeter of a rectangle whose area is 400 ft2.

5. Suppose you want to fence a rectangular area with one side against a barn (so no fencingneeds to be used for that side). If the amount of fencing to be used is 40 ft, find thedimensions of the rectangle that has the maximum area.

Section 13.1: Antiderivatives and Indefinite Integrals

1. What does it mean for F (x) to be an antiderivative of f(x)?

2. If n is any real number, what do the family of antiderivatives of xn look like?

3. Find an antiderivative of f(x) = 3x2.

4. Find an antiderivative of f(x) = 2x−1.

5. Find each indefinite integral.

(a)

∫ (x2 − 5

x+ 4ex

)dx

(b)

∫ (√x− 2

x2

)dx

(c)

∫ ( 1√x− x5

)dx

(d)

∫x(x2 + 1) dx

6. Circle true or false to the following statements.

(a) Every function has an infinite number of antiderivatives.

true false

(b) An antiderivative of f(x) = 1x

is F (x) = ln |x|+ 10.

true false

(c) An antiderivative of f(x) = 2x is x2 + x.

true false

(d) The constant function f(x) = 0 is an antiderivative of itself.

true false

9

(e) The function f(x) = 5ex is an antiderivative of itself.

true false

(f) If n is an integer, then1

n + 1xn+1 is an antiderivative of xn.

true false

Section 13.2: Integration by Substitution

1. Use integration by substitution to find each integral.

(a)

∫ √2− x dx

(b)

∫x√x2 + 1 dx

(c)

∫x2

√x3 + 5

dx

2. Integrate.

(a)

∫e−3x dx

(b)

∫x

x2 − 9dx

(c)

∫5t2(t3 + 4)−2 dt

Matched Problem 5 on p. 731, answer on p. 736

Section 13.4: The Definite Integral

1. Find the area under the graph of f(x) =√x but above the x-axis between x = 1 and x = 9.

2. Consider the graph of f(x) shown below, where A,B, and C represent the shaded regionsshown.

Suppose

area of A = 1.5 area of B = 2.5 area of C = 0.5

Find the following definite integrals.

(a)

∫ 0

−4f(x) dx

10

(b)

∫ 1

−4f(x) dx

(c)

∫ −4−5

f(x) dx

(d)

∫ 1

−5f(x) dx

(e)

∫ −50

f(x) dx

3. Find the following definite integrals.

(a)

∫ 2

1

(x3 − x

)dx

(b)

∫ 3

1

(6x2 − 1

x

)dx

(c)

∫ 1

0

x

x2 − 2dx

(d)

∫ 2

−1

√4− x dx

(e)

∫ 3

1

2x√x2 − 1

dx

Section 14.1: Area Between Curves

1. Find the area bounded by the graphs of f(x) = x and g(x) = −x2 + 4x as shown below.

11

2. Find the area between the graph of f(x) = x2−x−2 and the x-axis over [−2, 2]. The graphof f(x) has been graphed below. The area will be the sum of the areas of the two shadedregions A and B as shown.

Section 15.1: Functions of Several Variables

1. Given the functions f(x, y) = x2 − xy + 2y and g(x, y) =x2 − 2

y + 1, find

(a) f(0, 1)

(b) f(1, 2)

(c) g(2, 3)

(d) g(1, 2)

(e) g(2,−1)

(f) f(2, 2)− g(2, 2)

Section 15.2: Partial Derivatives

1. Given the functions f(x, y) = x2 − xy + 2y and g(x, y) =x2 − 2

x + 1, find the indicated partial

derivatives.

(a) fx

(b) fy

(c) fxx

(d) fyx

(e) gx

(f) gy

(g) gxy

(h) gyy

12