Math 1210
Practice for Midterms 1 and 2
April 17th, 2014
The following is a list of questions covering some of the sections compressed
between 1.1 and 3.5 (Midterms 1 and 2). I will use this as a base for the
review sessions next week: please try to think about as many exercises as
possible so that you can ask me questions.
Sections 1.1-1.5: Computation of limits
Check the review sheet for midterm 1 for a summary of the limit theorems (1.3) and basic
rules to compte limits of trigonometric functions (1.4) and limits at infinity (1.5). Midterm
1 and the first few quizzes contain a few examples of each.
Section 1.6: Continuity of functions
There are two kinds of questions I can (and will...) ask from this section:
(i) Study the continuity of a function: you have to determine over which intervals a given
function is continuous and, for every point of discontinuity (removable, jump of vertical
asymptote), find the corresponding limit(s).
(ii) Use the intermediate value theorem to show that an equation has a solution over a
given interval.
1. Study the continuity of the function f (x) =
|x−3|
.
x−3
2. Study the continuity of the function f (x) =
x3 −27
,
x−3
27
2 −1
3. Study the continuity of the function f (x) = sin xx+1
4. Study the continuity of the functions f (x) =
5. Study the continuity of the function f (x) =
√ 1
4−x2
t 6= 3
t=3
and f (x) =
√ 1
.
4+x2
x2 +|x−1|
√
3
u+1

x<0
 x,
x2 ,
0≤x≤1
6. Study the continuity of the function f (x) =

2 − x, x > 1
7. Study the continuity of the functions f (x) = [[x]], f (x) = [[x2 ]], f (x) = [[2 sin x]] and
f (x) = [[|x|]].
8. Find the
 values of a and b so that the following function is continuous everywhere
 x + 1, x < 1
ax + b, 1 ≤ x < 2
f (x) =

3x,
x≥2
9. Use the Intermediate Value Theorem to prove that the equation x3 + 3x − 2 = 0 has a
real solution between 0 and 1. Give an interval of length 14 which contains this solution.
10. Use the Intermediate Value Theorem to prove that the equation (cos t)t3 +6 sin5 t−3 = 0
has a real solution between 0 and 2π.
√
11. Use the Intermediate Value Theorem to prove that the equation x − cos x = 0 has a
real solution between 0 and π2 .
12. Consider the function f (x) =
over the interval [−2, 2]?
1
x−1
Does the equation f (x) = 0 have any real solution
Sections 2.1-2.6: Derivatives and rules to compute them
You must be comfortable with these sections. Given any function, you must be able to
compute its derivative.
(i) Section 2.1: you should be able to use the definition of the derivative as
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
in order to compute derivatives of simple functions.
(ii) Sections 2.2 and 2.3: basic rules and rules for finding derivatives of trigonometric
functions.
(iii) Sections 2.4 and 2.5: chain rule and higher order derivatives.
Sections 2.7: Implicit differentiation
Check the review sheet for midterm 2 for a summary of implicit functions and the implicit
differentiation method.
1. Let y(x) be the function defined implicitly by the equation xy + y 3 = 2. Find y 0 (x) and
y 00 (x) and express them as functions of x and y only.
2. Let y(x) be the function defined implicitly by the equation 2x2 y − 4y 3 = 4. Find y 0 (x)
and y 00 (x) and express them as functions of x and y only.
3. Let y(x) be the function defined implicitly by the equation x = sin(y 2 ) + 2y 3 . Find y 0 (x).
4. Let y(x) be the function defined implicitly by the equation xy + sin(xy) = 1. Find y 0 (x).
5. Let y(x) be the function defined implicitly by the equation cos(xy 2 ) = y 2 + x. Find
y 0 (x).
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6. Find the equation of the tangent line to the curve given implicitly by
y + cos(xy 2 ) + 3x2 = 4
at (1, 0).
7. Show that the normal line to x3 + y 3 = 3xy at
3 3
,
2 2
passes through the origin.
8. Show that the graphs of 2x2 + y 2 = 6 and y 2 = 4x intersect at right angles.
Sections 2.8: Related rates
These are exercises in which a time-dependent variable y = y(t) is given and in which
ones wants to know how fast is y varying at a given time t0 (so one needs to determine y 0 (t))
1. Water is pouring into a conical tank at the rate of 8 cubic feet per minute. If the height
of the tank is 12 feet and the radius of its circular opening is 6 feet, how fast is the water
level rising when the water is 4 feet deep? (see Example 2 on page 136).
2. As the sun sets behind a 120-foot building, the building’s shadow grows. How fast is
the shadow growing (in feet per second) when the sun’s rays make an angle of 45 ( π4
radians).
3. An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer.
If the constant speed of the plane is 400 miles per hour, how fast is the distance from
the observer increasing 45 seconds later?
4. Sand is falling from a pipe at the rate of 16 cubic feet pr second. If the falling sand forms
a conical pile on the ground whose altitude is always 41 of the diameter of the base, how
fast is the altitude increasing when the pile is 4 feet high?.
√
5. A particle P is moving along the graph of y = 4 − x2 , x ≥ 2, so that the x-coordinate
of P is increasing at the rate of 5 units per second. How fast is the y-coordinate of P
increasing when x = 3?
6. The vertex angle θ opposite to the base of an isosceles triangle with equal sides of length
1
100 centimeters is increasing at 10
radian per minute. How fast is the area of the triangle
increasing when the vertex angle measures π6 radians?
Sections 2.9: Differentials and approximations
We defined the derivative of a function f at a point x as the limit
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
This means that when h is very small we have
f 0 (x) ∼
f (x + h) − f (x)
h
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namely
f (x + h) ∼ f (x) + f 0 (x)h
If we denote h by dx, the above expression reads
f (x + dx) ∼ f (x) + f 0 (x)dx
which means that if we perturb x a small amount dx, we can approximate f (x + dx) by
solely looking at f (x) and f 0 (x).
(i) For the function f (x) =
√
x, the above expression reads
√
√
1
x + dx ∼ x + √ dx
2 x
so we can approximate
√
√
9.02 as follows
9.2 =
√
√
1
9 + 0.2 ' 9 + √ · 0.2 = 16.0333
2 9
(ii) Similarly, we can approximate
√
15.7 as follows
√
√
√
1
15.7 = 16 − 0.3 ' 16 + √ · (−0.3) = 15.9625
2 16
1.
(i) Draw a picture of a curve y = f (x) and its tangent line at a point (x0 , f (x0 )) and
label. Identify on the picture the point (x0 + dx, f (x0 + dx)) and label ∆y and dy
(you may look this up on page 143).
(ii) For the curve y = x4 + 2x find the values of ∆y and dy for x = 1 and dx = 1.
2. Approximate the volume of material in a spherical shell of inner radius 5 centimeters
and outer radius 5.125 centimeters.
3. The outside diameter of a thin spherical shell is 12 feet. If the shell is 0.3 inch thick, use
differentials to approximate the volume of the region interior to the shell
4. The angle between the two equal sides of an isosceles triangle measures θ = 0.53 radian.
The two equal sides are exactly a = 151 centimeters long.
(i) How much does the length of the third side vary if θ increases (resp. decreases) by
0.005? Hint: the length of the third side is L = 2a tan 2θ .
(ii) How much does the area of the triangle side vary if θ increases (resp. decreases) by
0.005? Hint: the area of the triangle is given by A = 21 a2 sin θ.
5. Suppose that f is a function satisfying f (1) = 10 and f 0 (1.02) = 12. Use this information
to approximate f (1.02).
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Sections 3.1 Absolute extreme values
These are the maximum and minimum values that a function attains in a given interval.
A continuous function f defined over a closed interval [a, b] always attains both its maximum
and minimum value over the interval.
1. THEOREM. If a point c in [a, b] is an extreme point (either the maximum or the
minimum) of a function f , then c must be a critical point, namely one of the following:
(i) An endpoint of the interval of definition of f : (a or b).
(ii) A singular point: namely a point c such that f 0 (c) does not exist.
(iii) A stationary point: namely, a point c such that f 0 (c) = 0.
2. Thus, the procedure for finding the extreme points of a function f defined over an
interval [a, b] is the following:
(i) Make a list containing all the critical points (endpoints, singular points and stationary points).
(ii) Evaluate the function at each of these points.
(iii) The largest of these values will the the maximum of f and the smallest will be
the minimum (there might be more than one point where a function attains the
maximum value).
1. Find the maximum and the minimum values of the function f (x) = −2x3 + 3x2 on
[− 21 , 2].
2. Find the minimum and maximum values of the function f (x) = x2/2 on [−1, 2].
3. Find the minimum and maximum values of the function f (x) = x + 2 cos x on [−π, 2π].
4. Find the minimum and maximum values of the function f (x) =
1
:
1+x2
(i) On [−π, 2π].
(ii) On (−∞, +∞).
5. Find the minimum and maximum values of the function f (x) = sin x − cos x on [0, π].
6. Find the minimum and maximum values of the function f (x) = |3x − 2| on [−1, 4].
7. Find the minimum and maximum values of the function f (x) = x2 sec x on [− π4 , π4 ].
Sections 3.2: Monotonicity and concavity
1. Determine the intervals where the following functions are increasing, decreasing, concave
up and concave down.
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(i) f (x) = x3 − 12x + 1.
(ii) f (x) =
(iii) f (x) =
x2
.
x2 +1
√
sin x.
(iv) f (x) = x2/3 (1 − x).
Sections 3.3 Local extreme values
1. For each of the following functions, determine which critial points give a local maximum
value and which give a local minimum value.
(i) f (x) = x3 − 3x.
(ii) f (x) =
(iii) f (x) =
2
√x
.
x2 +4
cos x
.
1+sin x
2. Find the global maximum and minimum values (if possible) for the following functions:
(i) f (x) = sin2 (2x) on [0, 2].
(ii) f (x) =
2x
x2 +4
on [0, ∞).
√
(iii) f (x) = 6 x − 4x on [0, ∞).
(iv) f (x) = |x2 − 1| on [−2, 2].
(v) f (x) = sin(x2 ) on [0, π].
3. The equation y 3 + 7y = x3 defines y as an implicit function of x.
1. Write a sentence explaining what the above sentence means.
2. Find the local extreme values of of y(x).
3. Find the inflection points of y(x).
Sections 3.4: Optimization problems
1. What number exceeds its square by the maximum amount?
2. Find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse
x2 y 2
+ 2 =1
a2
b
3. Of all right circular cylinders with a given surface area, find the one with the maximum
volume (the ends of the cylinder are assumed closed)
4. A rectangle has two corners on the x-axis and the other two on the parabola y = 12 − x2 ,
with y ≥ 0. What are the dimensions of the rectangle of this type with maximum area?
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5. A rectangle has two corners on the x-axis and the other two on the circle x2 + y 2 = r2 ,
with y ≥ 0. What are the dimensions of the rectangle of this type with maximum area?
6. A fence h feet high runs parallel to a tall building and w feet from it. Find the length
of the shortest ladder that will reach from the ground across the top of the fence to the
wall of the building? (see figure 23 on page 175).
7. A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius
r and vertex angle θ. Find r and θ if its area is a constant A and the perimeter is a
minimum.
√
2
8. Find the points P and Q on the curve y = x4 with 0 ≤ x ≤ 2 3, that are closest to and
farthest from the point (0, 4). Hint: the algebra is simpler if you consider the square of
the required distance rather than the distance itself.
9. Find the greatest volume that a right circular cylinder can have if it is inscribed in a
sphere os radius r.
10. Show that the rectangle with maximum perimeter that can be inscribed in a circle of
radius r is a square.
Sections 3.5: Graphing functions using calculus
Given a function f defined over a interval or over the whole real line (−∞, +∞), you can
graph it precisely by following the steps below. NOTE: In the exam I will be giving you a
simplified expression for f 0 (x) and f 0 (x), so that you can focus on the sign study.
1. Compute the value of the function at the endpoints of the interval of definition or
compute the limits of the function at those endpoints.
2. Determine the domain of f and classify the discontinuities (removable, jump or vertical
asymptote).
3. Compute f 0 (x) and study its sign: this tells you over which intervals the function is
increasing or decreasing, the points for which f 0 vanishes (stationary points) and the
points at which f 0 does not exist (singular points).
4. Plot the stationary points and the singular points.
5. Using the first derivative criterion, you can then establish whether or not the stationary
points and the singular points are local maximums or minimums.
6. Compute f 00 (x) and study its sign: this tells you over which intervals the function is
concave up or down and also the points for which f 00 vanishes. If the sign of f 00 changes
at the latter points or at the singular points, then those are inflection points.
7. Plot the inflection points.
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8. Use all the information you have gathered to graph the function f .
1. Consider the following functions.
1. f (x) =
2x3
.
(x−1)3
2
6x
00
The derivatives are given by f 0 (x) = − (x−1)
4 and f (x) =
12x+12x2
.
(x−1)5
2. f (x) = (x2 + 2)2
3. f (x) =
2x+1
x+4
3
4. f (x) = 2x − 9x2 + 12x
5. f (x) =
3(x+2)2
.
(x−1)2
The derivatives are given by f 0 (x) = − 18(x+2)
and f 00 (x) =
(x−1)3
(x−1)(x−3)
.
(x+1)(x−2)
(x−1)(−x2 +4x−7)
6 (x+1)3 (x−2)3
6. f (x) =
The derivatives are given by f 0 (x) =
3x2 −10x+11
(x+1)2 (x−2)2
18(2x+7)
.
(x−1)4
and f 00 (x) =
For each of the functions above, answer the following questions:
(i) What is the domain of f ? Describe the discontinuities at the x-values for which f
is not defined.
(ii) Compute the limits limx→+∞ f (x) and limx→−∞ f (x).
(iii) Over which intervals is f increasing (resp. decreasing)?
(iv) What are, if any, the local extreme values of f ?
(v) Over which intervals is f concave up (resp. down)?
(vi) What are, if any, the inflection points of f ?
(vii) Sketch the graph of f .
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