201-SN2
DIFFERENTIAL CALCULUS
SCIENCE
REVIEW PACKAGE
The present document and the contents thereof are the
property and copyright of the Department of Mathematics,
Marianopolis College. No part of the present document may
be sold, reproduced, republished or redisseminated in any
manner or form without the prior written permission of the
chair of the Department of Mathematics.
Contents:
-
Review problems.
Short answers to review problems (with occasional hints).
One past final exam.
Short answers to the final exam.
Note:
One past final exam is provided as practice. It should not be taken as an indicator of
the content of any future final exam.
1. Make a simple sketch of the following functions; state the domain, range, x- and y-intercepts, if any.

sin(4 x)
 x + 1 if x  0,
a) f ( x) =  x
b) g ( x) =
2

 e − 3 if x  0.
2. a) Let f ( x) = 3 x − 1 and g ( x) = x3 + 1 .
Find a simplified expression for ( g  f )(x) and for ( f  g )(x) .
b) Based on your findings in a), what can you say about those functions f (x) and g (x) ?
ln x + 3
c) Find the inverse of the function r ( x) =
1 − 2ln x
3. a) Solve for 𝑥, giving the exact solutions satisfying −𝜋 < 𝑥 ≤ 𝜋 : sin( x) − sin(2 x) = 0 .
2𝜋
b) Find the exact value of arctan (tan 3 ).
4. Simplify the following expressions as algebraic expressions.
1
a) cos ( tan −1 a3 )
b) tan(arcsin √𝑥 + 1)
c) cos (2 sec −1 (𝑥))
5. Evaluate the following limits.
x −2 − 14
x 2 + 13x + 36
a) lim
b) lim
x →−4 3x 2 + 14 x + 8
x →2 x − 2
1− 2x − x +1
x + 6 − 5x − 6
d) lim−
e) lim
x
→
3
x →0
6 x − 18
x
3 
 1
c) lim+ 
− 3 
x →1  x − 1
x −1 
f)
x →
x +2
3x + 10
2
g)
lim
x → −
6. Evaluate the following limits.
cos( x 2 )
a) lim
b) lim+ x sin 1
x
x → 1 + x 2
x →0
( )
7. Find all horizontal asymptotes for h( x) = x + x 2 − 2 x .
8. Consider the function f ( x) =
6 − 9 x + 3x 2
.
− x2 + 4 x − 3
a) Compute lim f ( x) ;
x →1
b) compute lim f ( x) and lim f ( x) ;
x →
x →−
c) compute lim+ f ( x) and lim− f ( x) ;
x→3
lim e − x ln x
x→3
d) illustrate the results obtained in a), b), and c) on a simple sketch.
9. Evaluate the following limits.
a) lim arctan
x→ 0
d)† lim
x→ 
†
( )
1
x
2 𝑥
b) lim+ t (ln t )2
c) lim (1 − 𝑥)
e) lim+ (sin x) x
f)
𝑥→∞
t →0
−1
tan (sin x)
x
x →0
1 
 1
lim 
−

2x 
 sin x
x → 0+
Recall that tan −1 x = arctan x .
10. Find constants a and b such that lim
𝑥→0
√𝑎𝑥+2𝑏−1
=2.
𝑥
9 sin(𝑥)
if 𝑥 < 0,
𝑥
11. Consider the function 𝑓(𝑥) = { 3 − 2𝑘
(𝑥 + 𝑘)
2
if 𝑥 = 0,
if 𝑥 > 0.
a) For what value(s) of 𝑘, if any, does lim f ( x) exist?
x →0
b) For what value(s) of 𝑘, if any, is 𝑓(𝑥) continuous at 𝑥 = 0?
( 4 ) + ln ( x − 4x − 5) is continuous.
12. Find the interval on which the function f ( x) = arcsin x
2
10
13. a) Show that the equation 2𝑥 = 𝑥 must have a solution ‘c’ somewhere in the interval (0,∞).
b) Suppose the domain of some function g is (0,∞) and g(c) = 2 where ‘c’ is the same as in part a).
Is the following true or false?
10
lim (2𝑥 − 𝑥 + 𝑔(𝑥)) = 2 .
𝑥→𝑐
3
14. Consider the function 𝑓(𝑥) = {
√1+𝑘𝑥−1
, 𝑥 ≠ 0;
𝑥
5,
𝑥 = 0.
For what value(s) of k is the function continuous at 𝑥 = 0?
1
15. Let 𝑓(𝑥) = 𝑥 2. Using the limit definition, find the slope of the tangent line to f at x = 2 .
16. Below, you are given a graph of the position of an object (in m) vs time (in s).
a) What is the average velocity of the object over the
first two seconds?
b) Estimate the instantaneous velocity of the object at
time 𝑡 = 1 s.
17. A ball is dropped from a tower starting from a state of rest at time 𝑡 = 0; its position in meters is
given by 𝑦(𝑡) = 70 − 4.9 𝑡 2 where t is measured in seconds.
a) How much distance does the ball travel during the time interval [2,2.5].
b) Compute the average velocity over [2,2.5].
c) Approximate the ball’s instantaneous velocity at 𝑡 = 2 by calculating the average velocity on the
shorter time interval [2,2.05].
d) Use a limit to obtain the instantaneous velocity at 𝑡 = 2 and compare with the previous
approximation.
2x −1
.
x −3
a) Find the average rate of change of the function 𝑓 on the interval [1,1.5],
find the average rate of change of the function 𝑓 on the interval [1,1.02].
b) Use a limit to find the instantaneous rate of change of 𝑓 at 𝑥 = 1.
c) Find an equation of the line normal to the curve 𝑦 = 𝑓(𝑥) at the point where 𝑥 = 1.
18. Consider the function f ( x) =
3x 2 + 2 x + 1, x  0,
19. Let f ( x) = 
x  0.
 ax + b,
a) Using the definition of continuity, find which values of a and b make f continuous at x=0.
b) Using the definition of differentiability, find a and b so that f is differentiable at x=0.
20. Find the derivative of the following functions.
1
x3
+ 2 − log 3 x
a) f ( x) =
b) 𝑟(𝑥) = 𝑒 √𝑥+𝑥 + 𝑒 𝜋 − 𝑥 𝜋
3 2
x
cos( 2 )
tan x
d) f ( x) = 2
e) r ( ) =

x +1
g) 𝑔(𝑦) = sec(2 − √1 + 𝑦 2 )
j)
h)
f ( x) = arctan(7 x + 1)
f ( x) = x sin −1 (ln x) ( Recall that sin −1 x = arcsin x )
c) y = −2 x 3 x3 + 8
f) h(t ) =
i)
e−t + ecos(t )
t ln(t )
f ( x) = sin 4 (−e3 x )
21. Let f ( x) = sin(2x) + ax + b .
Suppose that the tangent line to y = f ( x) at x =  is given by y = 3x + 2 . Find a and b.
22. Differentiate the following.
a) y = ( ln(3x + 4) )
5
d) y = ex/3 csc( x4 )
b) y = cot 2 (  )
c) 𝑦 =
e) 𝑦 = arctan (𝑥 2 𝑒 𝑥 )
f)
arcsin(𝑥)
1+𝑥 2
y = ( tan 5 ( x) + 1)
2
23. a) Suppose that 𝐹(𝑥) = √𝑓(𝑔(𝑥)) , 𝑔(1) = 2, 𝑔′ (1) = 3, 𝑓(2) = 4, and 𝑓 ′ (2) = 5.
Evaluate 𝐹 ′ (1).
2
b) Suppose that 𝑦 = 𝑓(𝑥) is differentiable, that 𝑓(𝑔(𝑥)) = 𝑥 and that 𝑓 ′ (𝑥) = 1 + (𝑓(𝑥))
for all real x. Find and simplify 𝑔′ (𝑥).
dy
df
24. Use appropriate techniques to find
or
if
dx
dx
a) 𝑥 2 𝑦 3 − ln 𝑦 + 2𝑥 = 6
b) y = (cos x )
x
2
2𝑥+1 𝑥
c) 𝑓(𝑥) = ( 1−𝑥 )
25 The equation 𝑥 2 − 𝑥𝑦 + 𝑦 2 = 3 represents a rotated ellipse, that is, an ellipse whose axes are not
parallel to the coordinate axes. Find the points at which this ellipse crosses the 𝑥-axis and show that
the tangent lines at these points are parallel.
26. Consider the curve given implicitly by e( x −1) y = 2 x y 2 − 7 .
a) Find dy dx .
b) Find the tangent line(s) to the curve at x = 1 .
27. a) Use logarithmic differentiation to find f '(0) if f ( x) = ( x + 1)(2 x + 1)(3 x + 1)...(10 x + 1) .
b) If h( x) = 2 x1−1 , compute h ', h '', h ''' and then state the general form of the 𝑛𝑡ℎ derivative of h(x).
28. Anna and Jane both ran the Montreal marathon this year.
Their position according to time is provided in the graph on the right.
We compare their progress in the race.
a) Which runner had the greatest average velocity over the whole race?
b) Which runner finished first?
c) At what time is Jane running the fastest?
At what time is Anna running the fastest?
d) At some moment, Anna and Jane run at the same speed.
Using the graph, find an approximate value for that time.
29. a) The radius of a growing spherical cell is measured in micrometers (1𝜇m = 10−6 m).
Find the average rate of change of its volume with respect to 𝑟 when 𝑟 changes from 5 to 5.1 𝜇m.
b) Find the instantaneous rate of change of 𝑉 with respect to 𝑟 when 𝑟 = 5𝜇m.
30. Use differentials or a tangent line approximation to approximate the following.
1
a) 5 1.05
b) e−0.013
c) 102
31. Consider the function 𝑓(𝑥) = √𝑥.
a) Find 𝑇3 (𝑥), the degree 3 Taylor polynomial centered
at 𝑥 = 1 for the function 𝑓(𝑥).
b) Use 𝑇3 (𝑥) to approximate the value of √1.025.
c) What is the size of the error made when using 𝑇3 (𝑥)
to do this approximation?
d) Without any further computation, find the expression
of 𝑇1 (𝑥) centered at 𝑥 = 1 for 𝑓(𝑥) and graph it on
the given image.
32. Consider the curve described by the implicit equation x 3 + y 3 = 6 xy .
a) Find 𝑦′ and 𝑦′′ at point (3,3) .
b) Find the second order Taylor polynomial that best estimates the curve near 𝐴(3,3).
Use it to estimate the y-coordinate of a point on the curve at 𝑥 = 3.1 .
33. a) Show that the Mean Value Theorem fails for the function 𝑓(𝑥) = 𝑥 2/3 on the interval [-8,27]
and explain why.
b) Prove that the following equation is an identity on the interval x  (0,1) :
𝜋
arcsin(𝑥) + arcsin (√1 − 𝑥 2 ) =
2
34. Below you are given the graph of the derivative 𝑓′(𝑥) of a continuous function 𝑓(𝑥).
a) Find open intervals on which f is strictly
increasing / decreasing / concave up / concave down.
b) Give the sketch of a possible graph for 𝑓(𝑥)
such that 𝑓(2) = 0.
(x +1)2
4x 3 -12x
2(1- x 2 )
,
given
that
and
.
f
f
''(x)
'(x)
=
=
(x 2 +1)2
(x 2 +1)3
x 2 +1
Your sketch should consider the following:
domain, x- and y-intercepts, asymptotes, increasing/decreasing intervals, concavity.
35. Sketch the graph of f (x) =
Identify all local maximum and minimum points and points of inflection on the graph.
36. Find all relevant properties of the function f ( x) =
3( x − 2)2
and thereby provide a reasonable sketch
( x − 1)2
of its graph.
37.Show that the equation 2 x − 1 = sin x has exactly one solution.
( )
38. a) Consider f (x ) = arctan x8 . Find the absolute extrema on interval − 10,1 .
b) For which value(s) of "a" does g ( x) = ax3 + a2 x2 − 72x + 12 have a local minimum at x = 2 ?
c) Show that the function ℎ(𝑥) = ln (cos 𝑥) − 𝑥 4 − 6𝑥 + 10 is concave down on the interval
I = (− 2 , 2 ) .
39. A cylinder is to have a volume of 20π m3. The material for the top and bottom costs $10/m2 and
material for the side costs $8/m2. Find the radius of the cylinder that will minimize the building
costs.
40. Find the point(s) on x + 1 = − 12 y 2 that are closest to (− 4, 0 ) .
41. Show that among all rectangles with base on the x-axis and one corner at a point P( x, y) with x  0
on the curve y = e − x , the one with the largest area occurs when the point P is at a point of inflection
of the curve.
2
42. A rain gutter is to be constructed from a sheet of metal of width 30 cm bending up one third of the
sheet on each side through an angle of 𝜃 . Find 𝜃 so the gutter will carry the maximum amount of
water.
x


10 cm
43. A conical glass with height 10 cm and radius 4 cm is filled with water. Assume that water is leaking
from the bottom of the glass at a rate of 2 cm3 /s. Give the rate of change of the level of water in the
glass when this level is 5 cm. Remember that the volume of a cone of height ℎ and radius 𝑟 is
𝑉=
π r2 ℎ
3
.
𝜋𝑥
1
44. A particle is travelling along the curve 𝑦 = 2 sin ( 2 ). As it passes through the point (3 , 1), its
𝑥 −coordinate is growing at a rate of √10 cm/s. At this exact moment, what is the rate of change of
the particle’s distance to the origin?
45. Adam is painting a graffiti from the end of a 10m ladder that is leaned against a wall. Arthur, his
partner in crime, was meant to steady the ladder at its base, but he was distracted for a moment. The
base of the ladder thus began to slip. Despite his desperate situation, Adam had time to figure out
that he was dropping at a rate of 0.2 m/s at the instant he was 8m above the ground.
a) Find the rate at which the base of the ladder is slipping at this
same instant.
b) Is the area between the wall, the ladder and the ground growing at
this same instant? Answer by finding its rate of change.
c) Find the rate of change of the angle between the ladder and the
ground
SHORT ANSWERS TO REVIEW PROBLEMS
1: Graphs below.
a) Translate y = x left, translate y = ex down.
Dom: (−,0)  (0, ) , Ran: (−2, ) , x − ints : −1 & ln 3, y − int : none.
b) Divide period of y = sin( x) by 4, divide amplitude by 2.
Dom: R, Ran: [− 12 , 12 ] , y − int : 0, x − ints : 0,   ,   ,  3 ,   ,  5 , . . . or { k , k  Z }
4
a)
2
4
4
4
b)
x−3
2: a) i) x
ii) x
b) f and g are inverse functions.
c) r −1 ( x) = e1+2 x
3: a) Using the double angle identity for sine, we obtain x {0,  ,  3 } .
b) − 
3
3
4: a)
b)
a +9
2
5: a) -½
b) -¼
( x + 1)
c) 1
6: a) 0 (Use Squeeze)
−x
d) 1
c) 2 x 2 − 1 (Use double angle identity for cosine.)
e) − 19
f) 0
g) − 13
b) 0 (Use Squeeze)
7: None on the right, y = 1 on the left.
8: a) − 3 2
b) -3 and -3
c) -∞ and ∞
d) See on the right.
9: a)

b) 0
2
c)
1
e2
d) 0 (The arctangent function is bounded between two values… Use the Squeeze Theorem)
e) 1
f) +
10: b = 1 2
a=4
11: a) k = 3
b) k = −3
12: Function is continuous on its domain (why?). Get [-4,-1)
13: a) Define an appropriate continuous function and use the IVT with N = 0 .
b) F: g may not be continuous at c.
14: k = 15 (You may use l'Hospital's Rule.)
15: - ¼
b) Tangent line drawn at t = 1 has slope approximately 2 m/s also.
16: a) 2 m/s
17: a) 11.025 m
b) -22.05 m/s
c) -19.85 m/s
d) -19.60 m/s (from limit definition of derivative)
18: a) − 53  −1.667 , − 125
99  −1.263
5
b) − 4 = −1.25
13
c) y = 54 x − 10
19: a) lim+ f ( x) = lim− f ( x) requires b = 1 while a R.
x →0
x →0
b) With b = 1 , lim+
x →0
−2
20: a)
3
3 x
c)
f ( x) − f (0)
f ( x) − f (0)
= lim−
then requires a = 2 .
x →0
x
x
+ 3x 2 ln(2) 2 x −
3
5
1
x ln 3
−4 x 3 − 16
3
( x 3 + 8) 2
e) −2sin( ) −
2
cos( 2 )
2
(
) (
− y sec 2 − 1 + y 2 tan 2 − 1 + y 2
g)
i)
1+ y
2
−12e3x sin3 (−e3x )cos(−e3x )
21: a = 1, b = 2 + 2 .
)
b)
(1 + ) e
d)
sec2 x 2 x tan x
−
x 2 + 1 ( x 2 + 1) 2
1
2 x
x +x
−  x −1
−t
cost
e−t + ecos t sin t ( e + e ) (1 + ln t )
f) −
−
2
t ln t
( t ln t )
h)
7
1 + (7 x + 1) 2
j)
sin −1 ( ln x ) +
1
1 − (ln x)2
22: a)
c)
e)
23: a)
15 ( ln(3 x + 4) )
4
b)
3x + 4
1+𝑥2
−arcsin(𝑥)2𝑥
√1−𝑥2
f) 10 sec2 ( x) ( tan 9 ( x) + tan 4 ( x) )
1+𝑥 4 𝑒 2𝑥
4

d) e x /3 csc( x 4 ) ( 13 − 4 x3 cot( x 4 ) )
(1+𝑥 2 )2
𝑥
2𝑥𝑒 +𝑥 2 𝑒 𝑥
15
− cot(  ) csc 2 (  )
b) Differentiate both sides of f ( g ( x) ) = x ; obtain g '( x) = 1+1x2 .
dy 2 + 2 xy 3
=
24: a)
dx 1y − 3x 2 y 2
b)
dy
= (cos x) x ( ln(cos x) − x tan x )
dx
x2

df  2 x + 1  
3x 2
 2x +1 
c)
=
2
x
ln
+

 


dx  1 − x  
 1 − x  (2 x + 1)(1 − x) 
25: m = 2 at both x-intercepts.
y 2 − x y e( x −1) y
26: a) y ' =
x ( x − 1)e( x −1) y − 4 xy
b) At x = 1 , obtain y =  2 ; get lines y = − 14 x + 94 and y = 34 x − 114 .
27: a)
55
2
b) h '( x) = (−1)(2)(2 x −1)−2 ,
h ''( x) = (−1)(2)(−2)(2)(2x −1)−3 = ... ,
h '''( x) = ... = (−1)3 (1 2  3)(23 )(2 x −1)−4
We find h( n ) ( x) =
(−2)n n!
.
(2 x − 1)n+1
28: a) Both same average velocity.
b) Both finished at the same time.
c) Jane: at the start t = 0hr , Anna: at the end t = 4hr .
d) Must find a time at which the two tangent lines are parallel.
Diagram on the right: seems to occur at t = 2 hours .
29: a) Using Vsphere = 43  r 3 ,
b)
V
 320.5μm3 /μm
r
dV 4
= 3   3r 3−1  V '(5)  314.2μm2
dr
30: a) Differentials: 1 + 0.01 = 1.01 ;
b) Differentials: 1 + (−0.013) = 0.987 ;
c) Differentials: 0.01 + (−0.0002) = 0.0098 ;
Tangent line: 1 + 15 (1.05 −1) = 1.01 .
Tangent line: 1 + 1(−0.013 − 0) = 0.987 .
1
1
Tangent line: 100
+ (− 10000
)(102 −100) = 0.0098 .
31: a) T3 ( x) = 1 + 12 ( x − 1) − 18 ( x − 1)2 + 161 ( x − 1)3
b) 1.01242285625
c) Comparing to calculator value:
1.025 −1.01242285625  0.00000001968
d) T1 ( x) = 1 + 12 ( x −1) .
32: a) y =
y =
2 y − x2
at (3,3) : y ' = −1 .
y2 − 2x
( 2 y − 2 x ) ( y 2 − 2 x ) − ( 2 y − x 2 ) ( 2 y y  − 2 )
( y − 2x)
2
2
b) T2 ( x) = 3 − 1( x − 3) − 249 ( x − 3)2 ;
at (3,3) , using y ' = −1 : y '' = − 489 .
y  T2 (3.1)  2.87333 .
33: a) 'c' cannot be found: MVT may fail as f is not differentiable on the whole open interval (−8, 27) .
b) Put 𝑓(𝑥) = arcsin(𝑥) + arcsin(√1 − 𝑥 2 ), compute f '( x) and show that it is zero, invoke a
√2
result proven in class, finalize by finding the value of 𝑓(𝑥) at 𝑥 = 2 or any point of (0,1).
34: a) f is strictly increasing on (−, −1)  (1, 2) ,
strictly decreasing on (−1,1)  (2, ) ,
concave up on (0, 2) , and
concave down on (−,0) .
35:
−2 x )
36: f '( x) = 6(( xx−−1)2)3 , f ''( x) = 6(5
.
( x −1)4
Graph on the right.
b)
37: Rewrite in form f ( x) = 0 ;
use IVT to show at least one root,
use RT to show at most one root.
38: a) Max is arctan(108 ) at endpoint x = −10 ,
min is 0 at critical number x = 0 .
b) For x = 2 to be a critical number, we need a = 3 or a = −6 .
When a = 3 , g (2) is a local min by the second derivative test;
When a = −6 , g (2) is neither a max nor a min by the first derivative test.
c) Argue that h ''( x)  0 on I.
39: r = 2 m; justify min cost using first derivative test.
40: Points at minimum distance are (−3, 2) and (−3, −2) ;
verify minima using the first derivative test.
41: Max area when x = 1 2 ; justify max using first derivative test; verify ( 12 , 1e ) is an inflection point.
42:  =  3 ; justify max.
𝑑ℎ
1
43: 𝑑𝑡 = − 2𝜋 cm/s
3√3
44: 1 + 2 𝜋 cm / s
4
45: a) 15 𝑚/𝑠
7
b) Area is growing at a rate of 15 𝑚2 /𝑠
1
c) − 30 𝑟𝑎𝑑/𝑠
Reminder:
This past final exam is provided as practice. It should not be taken as an
indicator of the content of any future final exam.
201-SN2
DIFFERENTIAL CALCULUS
Final Examination
Instructions:
You have three hours to complete this exam.
Give exact answers ( 2 is exact, 1.414 is an approximation of
2 ) unless otherwise instructed.
Complete, justified answers are always required. Show all your work.
Only college approved calculators are allowed: Texas Instruments Models TI-30XII (B or S).
Problem 1.
4 pts
Consider the function f ( x) =
x + 6 x + 10
.
x2 + 5x + 4
2
(a) Determine whether x = −4 is a vertical asymptote of the graph of f ( x) . Compute all relevant limits.
(b) Give a sketch of the graph of f ( x) near x = −4 .
Problem 2.
4+4+3+5 pts
Evaluate the following limits or explain why they do not exist. Use the symbols  whenever
appropriate.
cos(2 x) − 1
(a) lim
x →3 (sin( x)) 2
Problem 3.
(b) lim
x →−
4 x 2 − 14 x
14 x + 1
x3 − 27
(c) lim
x →3 | x − 4 | −1
(d) lim+ (sin x)2 x
x →0
5+3 pts
Let f ( x) = 3x + 1 .
(a) Find the derivative of f ( x) using the limit definition. No points will be given for an answer using
the derivative rules.
(b) Find the point(s) ( x, y) on the graph of y = 3x + 1 where the tangent line is parallel to the line
given by 3x − 8 y = 5 .
Problem 4.
3+4 pts
 − x2 + 2
 2 x 2 + 2 if x  0,
Consider the function f ( x) = 
 1
if x  0.
 x + 1
(a) Find the equation(s) of the horizontal asymptote(s) of f ( x) , if any. Compute all relevant limits.
(b) Determine whether f ( x) is differentiable at 0 by computing the following two limits.
f ( x) − f (0)
f ( x) − f (0)
lim−
lim+
x →0
x →0
x−0
x−0
If f is not differentiable, describe the type of problem point at x = 0 (discontinuity, vertical tangent
line, ...)
Problem 5.
3 pts
You are given the following information about an unknown function f ( x) .
lim f ( x) = −3
f (0) = 1
lim− f ( x) = 2
lim+ f ( x) = −4
x →−
x →0
x →0
lim f ( x) = 5
x →+
Assume that you are also told that f '(4) = 0 . Give a sketch for the graph of f ( x) which fits with the
given information.
Problem 6.
4+4+4 pts
Find the derivative of the following functions. Do not simplify your answers.
2
cos x
(t 4 )
+ 3
(a) f ( x) = 5
(b) h(t ) = 3 sec(t )
(c) g ( x) = ( x) tan( x + x )
ln x
Problem 7.
3+3 pts
(a) Give the domain, range and draw the graph of arctan( x) .
d
1
(b) Prove that
.
( arctan x ) =
dx
1 + x2
Problem 8.
5 pts
A camera is mounted 3000 m from the base of a launching pad to follow the takeoff of a rocket. Assume
that when the rocket is 4000 m above the launching pad, it is rising vertically at 880 m/s. How fast must
the camera angle change at that instant to stay aimed at the rocket?
Problem 9.
Consider the curve 3xy + 25 = 4x2 + y3 .
(a) Find y ' .
(b) Compute y ' at the point P(2,3) .
(c) Which of the points on the diagram could be P ?
Briefly explain your choice.
4+1+1 pts
Problem 10.
7 pts
The Medieval company sells cylindrical barrels of wine at a price of $
20d where d is the length of the diagonal of the barrel in meters. Johann
Kepler will spend $200 for one barrel of wine. He will therefore buy a
barrel with a diagonal of d = 10 m, but he can otherwise choose the radius
and the height. To maximize the volume of wine, what radius and height
should his barrel have? How much wine would Johann then get? Show
that the value that you have found is indeed a maximum.
Problem 11.
6 pts
Consider the equation 3x + 2 x + 4 x = 10.
Prove that this equation has EXACTLY one solution by
7
3
(a) showing that it has at least one solution.
(b) showing that it has at most one solution.
Clearly name any theorem(s) that you are using.
Problem 12.
1
.
x3
(a) Give the third degree Taylor polynomial of f ( x) centred at x = 1 .
1
(b) Use the polynomial found in (a) to approximate
.
(0.9)3
Consider the function f ( x) =
Problem 13.
Please see next page.
3+2 pts
Problem 13.
Consider the function f ( x) = 3 x ( x − 7)2 . Assume that its first two derivatives are:
7( x − 1)( x − 7)
28( x + 1)( x − 3)
f '( x) =
f ''( x) =
2/3
3x
9 x5/3
Sketch the graph of f ( x) . Your work must consider the following.
(a) The domain of f ( x) .
(b) The x and y intercepts of f ( x) .
(c) The intervals on which f ( x) is increasing and decreasing. The local extrema ( x, y) .
(d) The point(s) ( x, y) , if any, where f ( x) is not differentiable. The type of problem point(s)
(discontinuity, vertical tangent line,...)
(e) The intervals on which f ( x) is concave up and concave down. The inflection points.
Fill in the "RECAP OF RESULTS" table below.
15 pts
SHORT ANSWERS TO PAST FINAL
lim f ( x) = + .
1: (a) Yes. lim+ f ( x) = − ,
x →−4−
x →−4
(b) − 17
2: (a) -2
3: (a)
3
2 3x + 1
(b)
(d) e0 = 1
(c) -27
(b) (5,4)
4: (a) HA on the right is y = 0 : lim f ( x) = 0 ;
x →
HA on the left is y = − : lim f ( x) = − 12 .
1
2
x → −
f ( x) − f (0)
f ( x) − f (0)
= −1 ;
= 0 , lim+
x →0
x →0
x
x
not differentiable: sharp corner.
(b) lim−
5:
6: (a)
1  cos x 


5  ln x 
−4/5
(− sin x)(ln x) − (cos x) 1x
+ 0
(ln x)2
(b) 3t ln(3)  4t 3 sec(t ) + 3t sec(t ) tan(t )
4
4
(c) x tan( x+ x ) (1 + 2 x)sec2 ( x + x 2 ) ln( x) + 1x tan( x + x 2 ) 
2
7: (a) Domain = R,
Range: (− 2 , 2 )
(b) ----8:
66
625
9: (a)
10: r =
rad/s
8x − 3 y
,
3x − 3 y 2
50
3
m, h =
(b) − 13
100
3
(c) D: only choice where slope is negative.

m, V = 500
m3 ; justify max by checking endpoints or using a derivative test.
3 3
11: (a) Rewrite as p( x) = 0 and use the IVT.
(b) Assume two real roots c1 and c2 for p( x) and obtain a contradiction using Rolle's Thm.
12: (a) T3 ( x) = 1 − 3( x − 1) + 6( x − 1)2 − 10( x − 1)3 .
(b) 1+0.3+0.06+0.01 = 1.3700
13: (a) Dom ( f ) = R
(b) x − int : x = 0 and x = 7 , y − int : y = 0 .
(c) Inc: (−,1)  (7, ) , dec: (1, 7) , local max: (1,36) , local min: (7,0) .
(d) (0,0) : vertical tangent.
(e) Concave up: (−1,0)  (3, ) , concave down: (−, −1)  (0,3) ,
Three IPs: (−1, −64) , (0, 0) , and (3,16 3 3)  (3, 23.1) .