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2 Limits and Continuity

2.6 Continuity

1. Let f be a function defined on the interval [a, b].

(a) Suppose f is continuous on [a, b]. Also, suppose f(a) > 0 and f(b) < 0. What must be true

about the function? What theorem justifies your answer?

(b) Is your answer to part (a) true if f is not continuous on [a, b]? Explain or give a counterexample.

(c) Is your answer to part (a) true if we assume that f(a) < 0 and f(b) > 0? Explain or give a

counterexample.

2. Use your observations from Problem 1 to prove that the equation x

4 + x − 3 = 0 has a solution in

the interval (1, 2).

3. A mountain guide leaves the base camp at 8 a.m. and leads a group of hikers to the summit, where

they arrive at 8 p.m. The group cooks dinner, gets some sleep, and then leaves the summit at

8 a.m. the next day. They return along the same path and arrive at the base camp at 8 p.m. Is

there some point along the path that the guide crosses at exactly the same time on both days?

Explain.

4. True or False? At some time since you were born your height in inches was exactly equal to your

weight in pounds. Explain.

3 Derivatives

3.1 Introduction to the Derivative

1. Explain the difference between the two expressions

f(b) − f(a)

b − a

and lim

b→a

f(b) − f(a)

b − a

.

1

MA 123: Calculus 1 Discussion Worksheet Pack 4: Sections 2.6 and 3-1–3.3

2. Consider the function f(x) = x

2 − 4x.

(a) Sketch the graph of f.

(b) Without using calculus, sketch the line tangent to the graph of f at x = 2.

(c) What is the slope of this tangent line?

(d) Use the definition of the derivative to calculate f

0

(x).

(e) Evaluate f

0

(2) using your answer from part (d).

(f) How do your answers to parts (c) and (e) compare?

3. Suppose y = 5x + 1 is the line tangent to the graph of y = f(x) for some function f at the point

where x = 2. Calculate f(2) and f

0

(2).

4. Let f be the function f(x) = x

3

.

(a) Use the definition of the derivative to compute f

0

(x).

(b) Using your answer from part (a), evaluate the limit lim

h→0

(2 + h)

3 − 8

h

.

5. Let g(x) = 1/(2x).

(a) Use the definition of the derivative to compute g

0

(x).

(b) Use your answer from part (a) to find the slope of the line tangent to the graph of g at x = 1.

(c) Write an equation for this tangent line.

(d) Use your graphing calculator to plot the graph of y = 1/(2x) and your answer from part (c).

6. Each of the expressions below represents the derivative of some function f at a number a. Determine such an f and a. Do NOT evaluate the limit. (Hint: Refer to the definition of the derivative

or to Problem 1.)

(a) lim

h→0

√5

32 + h − 2

h

(b) limx→3

2

x − 8

x − 3

3.2 Working with Derivatives

1. The graph of a function h is shown to the

right. Use this graph to determine if h

is differentiable at each of the indicated

points. Justify your responses.

(a) x = −3

(b) x = −2

(c) x = −1

(d) x = 3

2

MA 123: Calculus 1 Discussion Worksheet Pack 4: Sections 2.6 and 3-1–3.3

2. Sketch the graph of a function f that satisfies all of the following conditions:

(i) f(0) = 0 (ii) f

0

(0) = −1 (iii) f

0

(2) = 0 (iv) f(3) = −2

3. The graph of a function f is shown on the left below.

(a) On the same figure, sketch the lines tangent to the graph at the points A, B, C, D, and E.

(b) Approximate the slope of each of the tangent lines you sketched in part (a).

(c) Make a rough sketch of the graph of the derivative f

0 on the axes provided on the right.

4. Given the graph of the function g shown on the left below, sketch the graph of the derivative

of g on the axes provided on the right. (Use open dots to indicate the places where g is not

differentiable.)

1 2 3 4 5 6 7

-2

-1

1

2

1 2 3 4 5 6 7

-2

2

3

MA 123: Calculus 1 Discussion Worksheet Pack 4: Sections 2.6 and 3-1–3.3

5. Consider the functions f whose derivative satisfies the conditions

f

0

(x) =

1 if x < 2,

0 if 2 < x < 4, and

−1 if x > 4.

(a) Sketch the graph of one such function f that is continuous on the interval [0, 6].

(b) Are there any other such functions that are continuous on the interval [0, 6]? Explain your

answer.

(c) Sketch the graph of another such function f that is not continuous on the interval [0, 6].

3.3 Rules of Differentiation

1. Let f(x) = 1 + √

4x.

(a) Use the definition of the derivative to calculate f

0

(x).

(b) Use the power rule to calculate f

0

(x).

2. Calculate the derivative of g(x) = (x

2 − 2)(3x − 1) by first simplifying the expression for g(x)

algebraically.

3. Calculate the derivative of f(x) = 3x

2 − 5x

x

2

by first simplifying the expression for f(x) algebraically.

4. Let f(x) = (

x

2 − 2x, if x ≤ 3; and

ax − b, if x > 3.

Find the values of a and b such that f is differentiable at x = 3.

5. True or False? Explain your answer.

(a) d

dx(x

5

) = 5x

4

(b) d

dx(e

5

) = 5e

4

(c) d

dx(e

5x

) = 5e

4x

A. Challenge Problems

1. Determine an equation for a function with a vertical asymptote at x = −2 , a removable discontinuity at x = 1, and a horizontal asymptote at y = 0.

2. Determine an equation for a function with a vertical asymptote at x = 1, a removable discontinuity

at x = −2 , and a horizontal asymptote at y = 5.

3. State the intervals of continuity for the function f(x) = r x

x − 2

.

4. Identify and classify the discontinuities of the functions

f(x) = |3x − 6|

x − 2

and g(x) = x

2 + 4x − 12

x

2 − 2x

.

4

MA 123: Calculus 1 Discussion Worksheet Pack 4: Sections 2.6 and 3-1–3.3

5. Does f(x) = x sin (1/x) or g(x) = sin (1/x) have a removable discontinuity at x = 0? Justify your

answer.

6. Consider the equation from Problem 2 of Section 2.6. Use a bisection method to approximate the

value of the root that lies between x = 1 and x = 2 to the nearest hundredth. (That is, cut

the interval (1, 2) in half and determine if the solution lies in (1, 1.5) or (1.5, 2), then repeat this

process.)

7. True or False? Explain. On the equator there are two diametrically opposed places that are

exactly the same temperature at the same time.

8. Suppose I is an interval on which a function f is defined and c is a number in I. If f(c) ≥ f(x)

for all x in I, then we say that f(c) is a local maximum value of f. Similarly, if f(c) ≤ f(x) for

all x in I, then we say that f(c) is a local minimum value of f. This definition is illustrated here:

(i) f(a) is a local maximum value because f(a) is greater than any other

output on I1.

(ii) f(b) is a local minimum value because f(b) is less than any other output on I2.

I1 I2

(a) If f is differentiable on the entire real line, what must be true about the line tangent to the

graph of f at a local maximum or minimum?

(b) What does this mean about the derivative of f at a local maximum or minimum?

9. Suppose I is an interval on which the function f is defined. We say that f is increasing on I if

f(x2) > f(x1) whenever x1 and x2 are in I and x2 > x1. Similarly, we say f is decreasing on I if

f(x2) < f(x1) whenever x1 and x2 are in I and x2 > x1 .

(a) If f is differentiable and increasing on an interval, what must be true of f

0

(x) on this interval?

(b) If f is differentiable and decreasing on an interval, what must be true of f

0

(x) on this interval?

(c) Suppose f is differentiable on the interval (a, c). If f is increasing on (a, b) and decreasing

on (b, c), what must be true about the function at x = b?

4 Applications of the Derivative

4.1 Maxima and Minima

23. Find the absolute maximum and minimum of the functions on the given interval and identify

where they occur.

(a) f(x) = −1/x2 on [1/2, 2] (b) f(t) = |t − 5| on [4, 7] (c) f(x) = e

−x

2

on [−2, 1]

4.2 What Derivatives Tell Us

1. (a) Suppose c is a critical point of f, what do know about f

0

(c)?

(b) How does knowing the critical points of a function help us understand its graph?

2. Determine whether the following statements are true or false. Provide an explanation or a counterexample.

(a) The function f(x) = √

x has an absolute maximum.

(b) The function f(x) = √

x has an absolute maximum on the interval [0,1].

3. Determine whether the following statements are true or false. Provide an explanation or a counterexample.

(a) If a function has an absolute maximum, the function must be continuous on a closed interval.

(b) If a function has the property f

0

(2) = 0, then f has a local minimum or local maximum at

x = 2.

(c) Absolute extreme values on an interval always occur at a critical point or an endpoint of the

interval.

(d) If a function has the property that f

0

(3) does not exist, then x = 3 is a critical point of f.

(e) If f is continuous on the closed interval [a, b] and the absolute maximum of f occurs at x = c

with a < c < b, then f

0

(c) = 0.

4. For the following functions, locate the critical values, intervals where the function is increasing/decreasing, local maxima and minima (if any), and absolute maxima and minima (if any).

f(x) = 2x − x

2

(a) on (−∞, 2] f(x) = √

25 − x

2

(b) on [−5, 5]

f(x) = x

2

4 − x

2

(c) on (−2, 1] f(x) = √

(d) 3 cos x + sin x on [0, 2π]

1

MA 123: Calculus 1 Discussion Worksheet Pack 7: Sections 4.1–4.2 and Some Review

Note: The following four-page document is a collection of 22 exercises that you can do to help you

prepare for our first midterm. Although these exercises address a wide range of topics that we have

studied so far this semester, they do not touch on every subject. As with every midterm and the final

exam, you are always responsible for all of the material that we cover in class as well as all of the

designated material from your text. The best way to study for our exams is to be sure that you are

very comfortable with the homework assignments, the worksheets, and the theory and examples that

we present in lecture.

There is a one-page document posted on MML that includes a paragraph about the material that

is covered on the first midterm. It is posted as an announcement.

4 Applications of the Derivative

4.4 Optimization Problems

1. A farmer has 120 meters of fencing with which he plans

to make a rectangular pig pen. The pen will have

one internal fence that runs parallel to the end fences

and divides the pen into two sections. Calculate the

dimensions that produce the pen of maximum area

assuming that the length of the larger section is to be

twice the length of the smaller section (see the figure). Justify the fact that your answer corresponds to the pen of maximum area.

2. A rectangle is inscribed in a right triangle with sides of length

6 cm, 8 cm, and 10 cm, respectively. Calculate the dimensions

of the rectangle of maximum area if two sides of the rectangle

lie along two sides of the triangle (see the figure). Justify the

fact that your answer corresponds to the rectangle of maximum

area.

3. An orchard presently has 25 trees per acre. The average yield has been calculated to be 495 apples

per tree. For each additional tree planted per acre, it has been predicted that the yield will decrease

by 15 apples per tree. Should additional trees be planted to increase the yield? If so, how many

should be planted to maximize the yield? Include a justification for your answer.

4. Calculate the point on the graph of y =

√

x that is nearest to the point (2, 0). Include a justification

that this point corresponds to a minimum.

1

MA 123: Calculus 1 Discussion Worksheet Pack 9: Sections 4.4–4.6

4.5 Linear Approximation

1. Consider the statement: “If the graph of y = g(x) is a line tangent to the graph y = f(x) at

the point x = a, then g gives a good approximation for f near x = a.” Do you agree with this

statement? Justify your response.

2. Let f(x) = x

2 and g(x) = 2x − 1.

(a) Confirm that the graph of g is tangent to the graph of f at the point (1, 1).

(b) Sketch the graphs of f and g over the interval −1 ≤ x ≤ 3.

(c) Note that g(x) can also be expressed in terms of the ∆x notation as g(x) = 1 + 2 ∆x where

∆x = x − 1 in this case.

(d) Use the equation g(x) of the tangent line or equivalently the ∆x notation to compute an

approximate value for f(1.06). Use a calculator or even multiplication by hand (!) to find

the exact value of f(1.06). How close is your approximation to the exact value?

(e) What is the maximum value of |f(x)−g(x)| on this interval? What is the significance of this

value in terms of the linear approximation?

3. (a) Use the equation of the tangent line to f(x) = √

√

x at a = 4 to approximate the value of

4.1.

(b) Use a calculator to calculate the error in this approximation (to the nearest ten thousandth).

4. Use linear approximation to estimate the value of e

0.03. Clearly show all work.

5. (a) Find the linearization at a = 0 of each of the following functions:

f(x) = (x − 1)2

g(x) = e

−2x h(x) = 1 + ln(1 − 2x)

What do you notice? Explain your observation.

(b) Graph each function and its linear approximation. (You may use a calculator or a computer

for this.)

(c) For which function is the linear approximation best/worst? Does the answer depend on the

chosen interval? Explain.

6. Suppose we know f(1) = 5 and the graph of y = f

0

(x) is as shown

on the right.

(a) Approximate the values f(0.9) and f(1.1).

(b) Can you tell if these estimates are too large or too small?

Explain.

2

MA 123: Calculus 1 Discussion Worksheet Pack 9: Sections 4.4–4.6

4.6 The Mean Value Theorem

1. (a) Before applying the Mean Value Theorem, what hypotheses must be checked?

(b) If the hypotheses of the Mean Value Theorem are satisfied, what conclusion can you make?

2. Use the graph of y = f(x) shown on the right to estimate

the values of c that satisfy the conclusion of the Mean

Value Theorem on the interval [0, 6].

3. True or False? For the function f(x) = x

2 on the interval

[−1, 2], there is a number c in the open interval (−1, 2)

such that

f

0

(c) = f(2) − f(−1)

2 − (−1) .

If true, compute all such numbers c. If false, explain.

4. True or False? For the function f(x) = |x| on the interval [1, 2], there is a number c in (−1, 2)

such that

f

0

(c) = f(2) − f(−1)

2 − (−1) .

If true, compute all such numbers c. If false, explain.

5. A pumpkin is dropped from a height of 64 feet. Its height s, in feet, is given by s(t) = −16t

2 + 64

where time t is given in seconds.

(a) Determine the interval on which the function is valid. In other words, when does the pumpkin

hit the ground?

(b) Apply the Mean Value Theorem to establish that there is a time in this interval when the

pumpkin is traveling downwards at a speed of exactly 32 ft/sec.

Challenge Problems

1. (a) Explain why the Mean Value Theorem does not apply to f(x) = tan x on [0, π].

(b) Explain why the Mean Value Theorem does not apply to g(x) = x

2/3 on [−1, 1].

2. Prove/Explain: If f is differentiable on (−∞,∞), then between any two x-intercepts of f there

exists at least one x-intercept of f

0

.

3. Two racers start a race at the same moment and finish in a tie. Which statement must be true?

(a) At some point during the race, the two racers were not tied.

(b) The racers’ speeds at the end of the race must have been exactly the same.

(c) The racers must have had the exact same speed at the exact same time during the race.

(d) There exist times t1 and t2 such that the first racer’s speed at t1 was equal to the second

racer’s speed at t2, but t1 6= t2.

4 Applications of the Derivative

4.2 What Derivatives Tell Us

1. (a) Sketch the graph of a differentiable function f that is increasing and concave up for all x.

What do we know about f

0 and f

00?

(b) Sketch the graph of a differentiable function f that is increasing and concave down for all x.

What do we know about f

0 and f

00?

(c) Sketch the graph of a differentiable function f that is decreasing and concave up for all x.

What do we know about f

0 and f

00?

(d) Sketch the graph of a differentiable function f that is decreasing and concave down for all x.

What do we know about f

0 and f

00?

2. Sketch the graph of a function f that satisfies all of the following criteria:

(i) f is differentiable for all x. (ii) f is increasing and concave up on (−∞, −2).

(iii) f is increasing and concave down on (−2, 0). (iv) f is decreasing and concave down on (0,2).

(v) f is decreasing and concave up on (2,4). (vi) f is decreasing and concave down (4,∞).

3. Provide an example of each of the following:

(a) f

0

(0) = 0 but f

0 does not change sign at x = 0.

(b) f

0

(0) = 0 but the second derivative test is inconclusive.

(c) f has an inflection point at x = 0.

(d) f

00(0) = 0 but there is no inflection point at x = 0.

1

MA 123: Calculus 1 Discussion Worksheet Pack 8: Sections 4.2–4.4

4. Determine whether the following statements are true or false. Provide an example or counterexample. (Rough graphs of y = f(x) are acceptable.)

(a) f

0

can remain constant while f

00 changes.

(b) f

00 can remain constant while f

0

changes.

(c) f

0 and f

00 can both remain constant.

(d) f

0 and f

00 can both change.

(e) f can only change concavity if f

0

(x) 6= 0.

(f) f can have a critical point without changing concavity.

5. (a) Explain how to apply the First Derivative Test.

(b) Explain how to apply the Second Derivative Test.

6. An apartment building has 100 rental units. The management company knows from experience

that all apartments will be occupied if they charge $800/month. A survey suggests that, on

average, one additional apartment will remain vacant for every $10 increase in rent. What rent

should the management company charge to maximize revenue?

7. A boat leaves a dock at 2:00 P.M. and travels due south at a speed of 20 km/hr. Another boat

has been heading due east at a speed of 15 km/hr and reaches the dock at 3:00 P.M. At what

time were the two boats closest together?

8. For each of the following functions f, locate its critical values, inflection points, intervals where f

is increasing/decreasing, intervals where its graph is concave up/down, local maxima and minima

(if any), and absolute maxima and minima (if any).

f(x) = x

3

(a) (x + 2) for all x (b) f(x) = sin x cos x on the interval [0, π]

f(x) = x

2/3

(c) (5/2 − x) for all x f(x) = x

2 − 4

2x

(d) on its domain

4.3 Graphing with Calculus

1. (a) How do you determine the x-intercepts of a function?

(b) How do you determine the y-intercepts of a function?

2. (a) How do you locate the discontinuities of a function?

(b) How do you classify discontinuities as infinite (vertical asymptotes) or removable (holes)?

3. How do you determine the domain of a function?

4. (a) How do you determine the end behavior of a function?

(b) How do you know whether a function has a horizontal asymptote?

(c) Describe the possible end behavior of a function that does not have a horizontal asymptote.

2

MA 123: Calculus 1 Discussion Worksheet Pack 8: Sections 4.2–4.4

5. Make a complete graph of

f(x) = x

2 − 1

x

2 − 4

.

Label all components of your graph including the x- and y-intercepts, the vertical and horizontal

asymptotes, the local extrema, and the inflection points.

6. Identify the local and absolute extrema, the inflection points, and the x- and y-intercepts for the

following three functions:

(a) f(x) = x + cos x on [0, 2π) g(x) = x

4 − 6x

2

(b) h(x) = e

−(x−1)2

(c)

7. Let f(x) = x

2

e

−x

.

(a) Find the critical points of f.

(b) Determine the intervals on which f is increasing/decreasing.

(c) Determine the intervals on which the graph of f is concave up/down.

(d) Determine any local maximum/minimum values of f.

(e) Sketch the graph of y = f(x).

8. Consider the function graphed below. It is defined on the interval [−3, 4]. Sketch the graph of its

derivative on the same set of axes. What is the derivative at x = 1? Why?

-3 -2 -1 1 2 3 4 x

-3

-2

-1

1

2

3

y

3

MA 123: Calculus 1 Discussion Worksheet Pack 8: Sections 4.2–4.4

9. The graph of the derivative, y = f

0

(x), of a function f is given below.

(a) Sketch the graph of y = f(x) on the same set of axes assuming that f(0) = 0. In other

words, assume that the graph of f passes through the origin.

(b) Why is it important to specify that f passes through the origin? How would your graph have

been different if you were told that f(0) = 1?

-4 -3 -2 -1 1 2 3 x

-2

-1

1

2

3

y

10. Using all of the techniques discussed in this section, graph the function f(x) = 5

e

x + 1

.

4.4 Optimization Problems

1. A pipeline is to be constructed between an island and a utility company on the mainland. The

island is 6 miles from shore and the utility company is 10 miles down the coast, as shown in the

diagram below.

(a) If it costs 1.25 times as much to construct the pipeline underwater as it does on land, find the

value of x that minimizes the cost of construction.

(b) If it costs 1.1 times as much to construct the pipeline underwater as it does on land, find the

value of x that minimizes the cost of construction.

(c) Explain how this example illustrates that it is necessary to check both the critical points and

the endpoints when optimizing a function.

4

MA 123: Calculus 1 Discussion Worksheet Pack 8: Sections 4.2–4.4

2. A factory is capable of producing 5,500 widgets per day.

(a) If the cost of producing x widgets per day is given by

C(x) = 25, 000 + 0.04x +

1, 000, 000

x

,

find the number of widgets that will minimize daily production costs.

(b) If the cost of producing x widgets per day is given by

C(x) = 25, 000 + 0.04x +

1, 440, 000

x

,

find the number of widgets that will minimize daily production costs.

(c) Explain how this example illustrates that it is necessary to consider the domain of the function

you wish to optimize.

A. Challenge Problems

1. A farmer has L feet of fence available to build an enclosure consisting of n adjacent rectangular

pens. The farmer will use the barn wall to form one side of the enclosure (as shown below). Show

that the maximum area will always be achieved by allotting half of the available fence for the

length and dividing the other half evenly among the widths.

2. Given an arbitrary line y = mx + b and an arbitrary point (p, q) not on the line, find the point on

the line that is closest to (p, q). Solve this question twice: first use only geometry and then use

calculus.

3 Differentiation

3.10 Inverse Trigonometric Functions and Their Derivatives

1. Explain the relationship between y = sin x and y = sin−1 x. Sketch the graphs of both of these

functions. State the domain and range of each function.

2. Explain the relationship between y = cos x and y = cos−1 x. Sketch the graphs of both of these

functions. State the domain and range of each function.

3. Explain the relationship between y = tan x and y = tan−1 x. Sketch the graphs of both of these

functions. State the domain and range of each function.

4. Explain the relationship between y = sec x and y = sec−1 x. Sketch the graphs of both of these

functions. State the domain and range of each function.

5. (a) Evaluate sin−1

−

1

2

.

(b) Solve the equation sin x = −

1

2

for all values of x.

(c) Compare your answers to parts (a) and (b).

6. Evaluate:

(a) sin

tan−1

√

3

3

(b) cos

sin−1

4

5

(c) sin

cos−1

−

12

13 (d) sin

tan−1

−

1

2

7. Evaluate in terms of x assuming that x > 0:

(a) sin

tan−1

(9/x)

(b) tan

cos−1

4 Applications of the Derivative

4.7 L’Hˆopital’s Rule

1. (a) Before you apply L’Hˆopital’s Rule, what hypotheses must you check?

(b) If the hypotheses of L’Hˆopital’s Rule are met, what conclusion(s) can you make?

2. (a) Simplify the fraction (x

2 − 9)/(x − 3). Then use the result to calculate

limx→3

x

2 − 9

x − 3

.

Note that this method of calculating the limit is the one we discussed at the start of the

semester.

(b) As an alternative to the computation you made in part (a), apply L’Hˆopital’s Rule to evaluate

the limit.

(c) Calculate the limit

limx→3

x

2 − 9

x + 3

.

What theorem(s) justify your calculation?

(d) Explain why L’Hˆopital’s Rule does not help us evaluate the limit in part (c).

3. Calculate the following limits.

(a) limx→1

1 − x

e

x − e

(b) limx→0

sin x

2

x

(c) limx→∞

e

6x

x

2

(d) limx→∞

ln x

√

x

4. What happens if you try to apply L’Hˆopital’s Rule to the limit

limx→∞

x

√

x

2 + 1

?

Use another method to evaluate this limit.

5. Explain how to convert a limit of the form ∞ · 0 to a limit of the form ∞

∞

or

0

0

.

1

MA 123: Calculus 1 Discussion Worksheet Pack 10: Sections 4.7 and 4.9

6. Consider the limit limx→∞

xe−x

.

(a) Show that this limit can be rewritten so that we obtain an indeterminate form of type ∞

∞

.

(b) Use L’Hˆopital’s Rule to evaluate the limit.

7. (a) State the indeterminate form of lim

x→0+

x

2

ln x.

(b) Rewrite the expression and apply L’Hˆopital’s Rule to evaluate the limit.

8. In parts (a)–(c), evaluate the limit limx→∞

f(x)g(x).

(a) Let f(x) = x

−1

and g(x) = x.

(b) Let f(x) = x

−1

and g(x) = x

2

.

(c) Let f(x) = x

−2

and g(x) = x.

(d) Use the results of parts (a)–(c) to explain why ∞ · 0 is an indeterminate form.

9. (a) Explain the procedures we use to evaluate limits of the form 00

, 1∞, or ∞0

.

(b) State the indeterminate form of the limit lim

x→0+

(1 + x)

1/x. Then evaluate the limit.

(c) State the indeterminate form of the limit limx→∞

(1 + x)

e−x

. Then evaluate the limit.

10. In parts (a)–(c), evaluate the limits.

limx→∞

x

2 − x

(a) limx→∞

xe1/x − x

(b) lim

x→π/2−

(c) (sec x − tan x)

(d) Use the results of parts (a)–(c) to explain why ∞ − ∞ is an indeterminate form.

11. (a) What does it mean to say that f(x) grows faster than g(x) as x → ∞?

(b) Use a calculator or computer to graph f(x) = √

x and g(x) = 1 + ln x on the same set of axes.

(c) Based on the graph, which function appears to grow faster as x → ∞?

(d) Use limits to compare the growth rates of f and g as x → ∞.

12. (a) Use a calculator or computer to graph f(x) = e

x

2x

and g(x) = x

2 on the same set of axes.

(b) Based on the graph, which function appears to grow faster as x → ∞?

(c) Use limits to compare the growth rates of f and g as x → ∞.

13. Evaluate the following limits. Clearly state the indeterminate forms involved and the results you

use. (Hint: In part (c), use the change of variables x = 1/t.)

(a) lim

x→0+

sin x ln x

(b) lim

x→0+

x ln

x

x + 1

(c) limx→∞

x ln

x

x + 1

3 Differentiation

3.4 Product and Quotient Rules

1. Use the Product Rule to compute the derivative of g(x) = (x

2 − 2)(3x − 1).

2. Use the Quotient Rule to find the derivative of f(x) = 3x

2 − 5x

x

2

.

3. Suppose q(x) = f(x)/g(x). Use the table

x f(x) f

0

(x) g(x) g

0

(x)

10 4 −1 4 −5

to calculate q

0

(10).

4. Suppose p(x) = f(x) g(x). Use the graph

to the right to calculate p

0

(−2).

5. Calculate d

dx 7

x

+ e

2x

4 + 3√3

x

2

. (You do not need to simplify your answer).

1

MA 123: Calculus 1 Discussion Worksheet Pack 5: Sections 3.4–3.9

6. The approximate number of bacteria in a lab experiment is given by N(t) where t is time in hours.

(a) What is the meaning of N(5)?

(b) What is the meaning of N0

(5)?

(c) Given an unlimited amount of space and nutrients for the bacteria, what would you expect

to be greater N0

(5) or N0

(10)?

(d) If the amount of space and nutrients for the bacteria were limited, would your answer to

part (c) change?

7. Complete the following rules for differentiation:

d

dx (a) (c · f(x)) d

dx (b) (f(x) + g(x)) d

dx (c) (f(x)g(x))

d

dx

f(x)

g(x)

(d) d

dx (e) (f(g(x))

Challenge Questions for Sections 3.3 and 3.4

1. Find a function f such that f

0

(x) = 3x

2 + 6x + 5. How many such functions f are there? Justify

your answer.

2. Find a function g(x) such that g

0

(x) = 1

x

2

+

√3 x and g(1) = 0.

3. Use the Quotient Rule to prove the Reciprocal Rule: If g is differentiable and nonzero, then

d

dx

1

g(x)

= −

g

0

(x)

(g(x))2

3.5 Derivatives of Trigonometric Functions

1. Derive an equation for the tangent line to the curve y = e

x

cos x at the point where x = 0.

2. Compute f

(50)(x) for (a) f(x) = sin x and (b) f(x) = xe−x

.

3. A 10-foot ladder rests against a vertical wall. Let θ be the angle formed by the top of the ladder

and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom

of the ladder slides away from the wall, how fast does x change with respect to θ when θ = π/3?

4. Evaluate limx→0

sin(3x)

x

.

2

MA 123: Calculus 1 Discussion Worksheet Pack 5: Sections 3.4–3.9

3.6 Derivatives as Rates of Change

Suppose the position s of an object moving horizontally after t seconds is given by a function f, that

is, s = f(t) where s is measured in feet, with s > 0 corresponding to positions to the right of the origin.

For each of the two functions f below:

(a) Graph the position function.

(b) Find and graph the velocity function. When is the object stationary, moving to the right, and

moving to the left?

(c) Determine the velocity and the acceleration of the object at t = 1.

(d) Determine the acceleration of the object when its velocity is zero.

(e) On what intervals is the speed increasing?

1. f(t) = t

2 − 4t, 0 ≤ t ≤ 5 2. f(t) = 2t

2 − 9t + 12, 0 ≤ t ≤ 3

3.7 Chain Rule

1. If h(x) = f(g(x)), g(−1) = 4, g

0

(−1) = 3, f(4) = 6, f

0

(4) = 5, and f

0

(−1) = 7, compute h

0

(−1).

2. Suppose j(x) = g(f(x)) and k(x) = f(g(x)).

Use the graph to the right to evaluate j

0

(4)

and k

0

(−4).

3. Evaluate (a) d

dx

tan(x

5

)

and (b) d

dx

tan5

(x)

.

4. Find all points on the graph of f(x) = 2 sin x − sin2 x where the tangent line is horizontal.

5. Evaluate (a) d

dx(1+e

2x

)(3x−6)5

and (b) d

dx

e

sin x

cos x

. (You do not need to simplify your answer.)

6. Evaluate d

dx sin2

(e

3x

).

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