1. It is known that 5x = e

x has a solution in the interval [0,1].

(a) In order to approximate a solution on this interval by Newton’s Method, howmust you rewrite the equation?

(b) Write down the first two approximations x1 and x2 to this solution using Newton’sMethod with an initial guess of x0 = 0. You may leave x2 unsimplified.

(c) Find the linearization of h(x) = e

x � 5x at a = 14 .

3

2. The parts of this problem are not related.

(a) Compute limx!1(1 + kx)1/x, where k is a positive constant.

(b) Calculate the limit limx!1

x

k�kx+k�1(x�1)2 , where k is a positive constant.

4

(c) Verify the identity d

dx

arcsec x = 1x

px

2�1for x > 0.

(d) Compute the most general antiderivative of f(x) = 14x

+e

�3x+3�cos x+11x5/2+17.

5

3. A rectangle with sides parallel to the x-axis and y-axis is inscribed inside the ellipsex

2

a

2 + y

2

b

2 = 1. Of all such rectangles, find the dimensions of the one with maximumarea. Hint: draw a picture!

6

4. Consider the definite integralR 3

�3 5xdx

(a) Graph the integrand on the interval [-3,3] and compute the integral by interpretingit in terms of areas.

(b) Using the fact thatR 1

�3 5xdx = �20, use properties of the integral to computeR 3

1 5xdx. You may check you answer geometrically.

7

(c) Compute the integral using the definiton (limit of Riemann sums) and right end-points as sample points. You may use the formulas

Pn

j=1 ↵ = ↵n for any constant↵ and

Pn

j=1 j = n(n + 1)/2.

8

5. Suppose f(x) is a function defined on all of R with the following properties:

• f

0 and f

00 exist for all x (i.e. graph of f has no corners)

• f(0) = �2, f(2) = 2, f(4) = 1

• f

0(0) = f

0(2) = f

0(4) = 0

• f

00(�1) = f

00(1) = f

00(3) = 0

• f

0(x) < 0 for x < 0 and 2 < x < 4

• f

0(x) > 0 for 0 < x < 2 and 4 < x

• f

00(x) < 0 for x < �1 and 1 < x < 3

• f

00(x) > 0 for �1 < x < 1 and 3 < x

(a) Find all the local maxima and minima inside the interval �2 < x < 5.

(b) Find all the inflection points.

(c) Sketch the part of the graph of y = f(x) that lies over the interval �2 x 5.

(d) True or False: f(x) has a horizontal asymptote as x! �1.

(e) True or False: f(x) has a horizontal asymptote as x!1.

9

6. The parts of this problem are not related.

(a) A spherical balloon is inflated with helium at a rate of 8 cubic metres per second.How fast is the surface area increasing when the balloon has a radius of 2 metres?

(b) Let f(x) = cos(⇡x

2). Use the Mean Value Theorem to find a number M suchthat

|f(b)� f(a)| M |b� a|

for all a, b satisfying 0 a b ⇡.

11

7. Consider the function y = f(x) given implicitly by 2y + sin(x + y) = 2x.

(a) Find the equation of the tangent line to the curve at the point (⇡

2 ,

⇡

2 ).

(b) True or False: f(x) is increasing for all x.

12

PART I

1. Calculate the following limits and derivatives

(a) limx!0

x sin1

x

, using the Squeeze Theorem.

(b)d

dx

h(arctanh x

2)p

x

2 + 1i

2

PART II

2. The parts of this problem are NOT related

(a) Find the linearization of h(x) = e

x � 5x at a = 14 .

(b) Compute the most general antiderivative of f(x) = 14x

+ e

�3x+3 � cos x +11x5/2 + 17 and check your answer.

(c) Consider the function f(x) = 4x � e

x. We know by the Intermediate ValueTheorem that f(x) = 0 has at least one root in the interval [0, 1]. Explainwhy f(x) = 0 has at most one root in [0, 1]. Use principles of calculus, MeanValue Theorem, etc.

3

3. Consider the function y = f(x) given implicitly by 2y + sin(x + y) = 2x.

(a) Find the equation of the tangent line to the curve at the point (⇡

2 ,

⇡

2 ).

(b) True or False: f(x) is increasing for all x.

4

4. A runner is jogging directly east at a steady rate of 18 ft/sec. A road runningperpendicular to the runner’s path is ahead of the runner. There is a bear on theroad who is south of the east-west path, and he is also running, at a steady rateof 25 ft/sec.

(a) Draw a diagram depicting the situation, carefully labeling all the variablequantities.

(b) How fast is the distance between the jogger and the bear decreasing when therunner is 17 feet from the north-south road and the bear is 40 feet from theeast-west path?

(c) Given the data in part (b), does the jogger meet his demise, that is, will thejogger and bear cross paths at the intersection of the two roads at the sametime?

5

5. The function f(x) defined for all x 6= 0 has second derivative f

0 0(x)

f

0 0(x) = 2x� 1

x

2.

(a) Determine at which x, if any, the graph of f(x) has inflection points.

(b) Given the value f

0(�1) = 0, find the first derivative of f(x).

(c) Determine at which x, if any, the function f(x) has a local extrema (mini-mum/maximum).

6

6. Use appropriate methods of calculus to find the exact values of the followinglimits. If the limits do not exist, state this and justify your answer.

(a) limx!0

cos 6x

cot 7x

(b) limx!1

x

k � kx + k � 1

(x� 1)2, where k is a positive constant.

(c) limx!0+

✓1

x

◆x

7

7. A local farm is constructing a 500-square-foot rectangular pen to hold horses andcows. One side of the pen will be along a barn, so only three sides need to befenced in. There will also need to be a length of fence dividing the pen intotwo components, perpendicular to the barn. This will separate the horses fromthe cows. What should the pen’s dimensions be to minimize the total length offencing?

(a) Draw a diagram and carefully label/identify all the variables involved.

(b) Write down a single-variable function which needs to be minimized and givethe domain of the function.

(c) At what value does the function take on a minimum? Justify your answerusing calculus.

8

1. Consider the function f(x) = e

x+e

�x

2 .

(a) Sketch the graphs of e

x and e

�x on the same plane.

(b) Determine the range of f using part (a)

(c) Find the inverse of f .

2

2. The parts of this problem are not related.

(a) What is the domain of ln�x�

px

2 � 1�?

(b) Sketch the graph of g(x) =p

4� x

2 and determine the range of g.

(c) Simplify cot (arcsin x).

3

3. Assumelimx!c

f(x) = L.

Using the ✏-� definition of a limit, prove that

limx!c

↵f(x) = ↵L,

for any real number ↵.

4

4. Evaluate the following limits or state why it does not exist. Show and justify all steps.

(a)

limx!0

sin 3x

tan 5x

(b)

limx!0

1� sec2x

x

2

(c)

limx!2

x

2 + x� 6

x� 2x + 5

(d)

limx!2

x

2 + x� 6

x

2 � 2x

(e)

limx!0

x sin1

x

(Hint: Squeeze Theorem)

5

5. Let A be a constant and define the function h(x) = x

3 + x� 1 + Ax(x� 1)(2x� 1)

(a) Show that h has a root in the interval [0, 1].

(b) By calculating h(1/3) and h(2/3), show that for A su�ciently large, h has at least3 roots in the interval [0, 1].

6

6. With A and B constants, a function f is defined by:

f(x) =

8<

:

1x

+ A, if x < �1|x|, if �1 x 11x

+ B, if 1 < x

(a) Find A and B so that f is continuous everywhere.

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

(c) At which points does f fail to be di↵erentiable? Explain the answer.

(d) True or False: A continuous function is di↵erentiable.

7

7. Di↵erentiate the following functions

(a) e

x

2 tan x csc(6x)

(b) sin 6x

x

2+x+5

(c)p

x

x+1

(d) e

�x(4x + 1000)

8

8. There is a line passing through (2, 0) which is tangent to the graph of y = 1x

2 in thefirst quadrant. Find the equation of this line.

9

9. Consider a particle moving along the x-axis, whose position is described by the functions(t) = 12p

t

, where position is measured in metres and t is seconds.

(a) Use the definition of the derivative to compute the instantaneous velocity of theparticle at time t = 3.

(b) Next, use any technique you wish to compute the acceleration of the particle att = 3.

(c) Is the particle moving forward or backward at t = 3? Is the particle speeding upor slowing down at t = 3?

10

Review Problems for exam 1

Note: these problems are in addition to the homework problems.

You can watch the solutions on the posted streaming power points.

1. Let

F (x) =

8><

>:

x

3+ A x ∑ 2

6x + 1 2 < x < 3

x

2+ 2 x ∏ 3

(a) Find the value of A that makes F (x) continuous at x = 2.

(b) For the constant A from (a): is F (x) diÆerentiable at x = 2?

(c) Is F (x) diÆerentiable at x = 3?

2. Show that the equation x

3= x

2+ 1 has at least one solution.

3. Use the definition of the derivative as a limit to find the derivative of f(x) =

1

x + 2

4. Find the center and the radius of the circle x

2+ y

2 ° 2x + 4y = 0

5. Evaluate each of the following limits.

(a) lim

x!3

px°

p3

x° 3

+ 3x

2(b) lim

x!0

tan (5x)

sin (3x)

(c) lim

x!3

|x° 3|x

2 ° 9

6. Find the derivatives of the following functions.

(a) F (x) =

se

x

x

2+ 3

(b) g(x) = 3 cos

4x · sin x

9(c) f(x) = e

sin x

(d) h(x) =

√x

2 ° ln x

3x + 2

!9

(e) F (x) =

tan x(x

2 ° 4x)

ln x

7. Let g(x) be a diÆerentiable function such that g(1) = 2, g(2) = 5, g(3) = 7,

g(4) = 2, g

0(1) = 3, g

0(2) = 2, g

0(3) = 8, g

0(4) = 10. Let f(x) = x

2+ x.

Find the exact value of: (a) (gf)

0(2) (b)

√f

g

!0

(3) (c) (g ± f)(1).

8. Solve log (5x) + log (x° 1)° 2 = 0

9. (a) Describe the rectilinear motion given at time t (in seconds) by s(t) = 20 + 8t° t

2(in

meters), for 0 ∑ t ∑ 10.

(b) Find the total distance traveled.

10. Find the domain of the function f(x) = ln (x

2 ° 4)

1

PART I

1. (a) Find the exact value of sin�⇡

3

�+ 2 cos

�⇡

3

�.

(b) If f(x) is the square of the distance from the point (2, 1) to a point (x, 3x+2)on the line y = 3x+2, then f(x) is a quadratic function, f(x) = Ax

2+Bx+C.Find A, B, and C.

2

PART II

2. Consider f(x) = 3x2+1x

2�9 .

(a) What is the domain of f?

(b) By evaluating relevant limits, determine the equations of all vertical asymp-totes of the graph of f(x). If there are none, say so and explain why not.

(c) By evaluating relevant limits, determine the equations of all horizontal asymp-totes of the graph of f(x). If there are none, say so and explain why not.

3

3. True or False. If true, prove the statement; if false, provide a counterexample.

(a) A continuous function is di↵erentiable.

(b) The following function is continuous at x = 1:

f(x) =

⇢3x+ 1, if x 1x

2+x�2x

4�x

3 , if x > 1

(c) The function r(x) = |x�17|17�x

is continuous at x = 17

4

4. Di↵erentiate the following functions

(a) e

x

2 tanx csc(6x)

(b) sin 6xx

2+x+5

(c)p

x

x+1

(d) e

�x(4x+ 1000)

5

5. The parts of this problem are related.

(a) Use the definition of derivative to find the derivative of f(x) = x�1x

.

(b) Find an equation of the tangent line to the graph of y = x�1x

at the pointwhere x = 2. In case you were unable to do part (a), you may use the factthat f 0(2) = 1

4 .

6

6. At time t, in seconds, the coordinate s(t), in feet, of a particle moving on a lineis given by

s(t) = t

2 � 8t+ 18.

(a) Find the particle’s average velocity over each of the time interval [4, 4.1]. Giveyour answer as a single number (you do not need to simplify) with appropriateunits.

(b) Calculate the particle’s average velocity over the time interval [4, 4 + 4t],where 4t > 0. Simplify your answer.

(c) Find the particle’s instantaneous velocity at t = 4. Use the meaning ofvelocity in terms of limits.

7

7. Let A be a constant and define the function h(x) = x

3+x�1+Ax(x�1)(2x�1)

(a) Show that h has a root in the interval [0, 1].

(b) By calculating h(1/3) and h(2/3), show that for A su�ciently large, h has atleast 3 roots in the interval [0, 1].

8

8. Let f(x) = 5x+ 4 so thatL = lim

x!2f(x) = 14.

(a) Draw a graph of f(x) and identify the "-neighbourhood of L = 14 for " = 0.01.

(b) For " = 0.01, find a corresponding value of � such that, for all x 6= 2:

if 2� � < x < 2 + �, then L� " < f(x) < L+ ".

Although you may use your graph as a guide, do this algebraically.

(c) Prove, using the "-� definition of a limit:

limx!2

5x+ 4 = 14.

9

Review Problems for final exam

Note:

1. The material covered in this set of problems doesn’t contain the material for the first andthe second exam. You should look for the sets of problems and the streaming power pointsfor the first and the second exams.

2. These problems are an addition to the homework problems.

3. You can watch the solutions on the posted streaming power points.

1. A continuous function f(x) is such thatZ 3

0f(x)dx = °4 and

Z 0

5f(x)dx = °10.

FindZ 5

3f(x)dx.

2. Find

(a)Z

x

2e

x

3+4dx (b)

Zsin x cos xdx (c)

Z 1

4° x

dx (d)Z

t

2

t° 1dt

3. Evaluate

(a)Z 4

1

1° xpx

dx (b)Z ln 2

0

e

x

e

x + 2dx (c)

Z 2

1

px(x3 °

px +

5px

)dx

4. Approximate the area under the curve y = x

2 + 2x ° 1, above the x-axis and betweenx = 1 and x = 4 using a Riemann sum with n = 3 and left hand endpoint of each of thesubintervals.

5. Let f(x) =Z

x

5t

3 sin (et)dt. (a) Findd

dx

f(x). (b) Evaluate f(5).

6. A farmer can get $3 per bushel of apples on September 15th. If he sell after that theprice drops 10 cents per bushel per day. On September 15th, the farmer has 200 bushels ofapple on the trees and the corps is increasing at a rate of 2 bushels per day. When shouldthe farmer pick the apples to maximize revenue?

1

Final Exam

r

R

θ

h

b

1 2-1-2

1

2

3y

x

S

4H

4