Topics and Concepts
1. Limits
(a) Evaluating limits (Know: limit exists if and only if the limit from the left is the same as
the limit from the right)
(b) Infinite limits (give rise to vertical asymptotes)
(c) Limits at infinity (give rise to horizontal asymptotes)
(d) Continuity (Know: limit definition, types of discontinuities)
2. Differentiation
(a) Limit definition and geometric reason for it
(b) Differentiation formulas (product rule, quotient rule, chain rule)
(c) Higher derivatives
(d) Implicit differentiation
(e) Finding equations of tangent lines
(f) Reasons a function might fail to be differentiable (Know: differentiable implies continuous, but not the other way around)
(g) Position, velocity, acceleration
3. Extreme Values
(a) Statement of the extreme value theorem
(b) Local vs Absolute maximum or minimum
(c) Definition of a critical point and classifying it
(d) Definition of concavity (using second derivative) and an inflection point
(e) Shape of a graph as determined by the first and second derivatives
4. Integration
(a) Definition of the integral via Riemann sums
(b) Antiderivative (and definition of indefinite integral)
(c) Fundamental Theorem of Calculus I and II
(d) Substitution
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Review problems on limits and derivatives
1. Evaluate the following limits.
x2 + ex
x→∞ 5ex + x10
(x + 1)2 − 1
x→0
x
√
4x + 1 − 3
(b) lim
x→2
x−2
1
1
(c) lim
− 2
x→0 x
x +x
(d) lim
(a) lim
(e) lim x2 e−x
2
x→∞
(f) lim x2 ln x
x→0+
√
(g) lim x
x
x→0+
1
. Say whether the function approaches positive
−1
or negative infinity from the right/left of the asymptote.
√
4x2 + 1
. State which asymptote f approaches as
3. Find the horizontal asymptotes of f (x) =
x−5
x → ∞ and as x → −∞.
2. Find the vertical asymptotes of f (x) =
x2
sin x
= 0.
x→∞ x2
4. Use the squeeze theorem to show that lim
5. Find a and b such that f (x) is continuous, where

2

−x + 3 if x < 0
f (x) = ax + b
if 0 ≤ x ≤ 2

 3
x −3
if x > 2.
6. The graph of a function f (x) is shown below.
(a) List the values of x at which f is discontinuous.
(b) List the values of x at which f is not differentiable.
p
7. Find f 0 (x) if f (x) = sin(x4 ).
8. Find y 0 implicitly if cos(xy) = 1 + sin(y). You do not need to solve for it.
9. Find the equation of the tangent line to the curve x2 + 2xy − y 2 + x = 2 at the point (1, 2).
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10. Find the absolute minimum and absolute maximum values of f (x) = (x2 − 1)3 on [−1, 2].
11. Find and classify the critical points of the curve f (x) = x4 − 2x2 + 2. Where is f increasing?
Decreasing? Where is the f concave up? Concave down?
12. The graph of f (x) is shown below.
(a) On what interval(s) is f 0 (x) positive? Explain.
(b) On what interval(s) is f 00 (x) positive? Explain.
13. If F (x) is a function such that F 0 (x) = 2 cos x and F (0) = 3, what is F (x)?
Z x
d
et sec t dt
14. What is
dx 2
Z x3
d
15. What is
sin(t2 ) dt
dx 0
Integration Practice: Chapter 5 Review, odd problems #9 - 19, #23 - 35
Concept Questions
Disclaimer: If I ask for a definition or what something “means” or “says” I am looking for the
mathematical definition/description. A description in words will only get partial credit.
1. Suppose that the graph of f is given. Write an equation for each of the graphs that are
obtained from the graph of f as follows.
(a) Shift 2 units upwards.
(b) Shift 3 units to the left.
(c) Reflect about the x-axis.
(d) Stretch vertically by a factor of 5.
(e) Shrink horizontally by a factor of 2.
2. How arcsin x defined? What is its domain and range?
3. What does the Squeeze Theorem say?
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4. What does it mean for f to be continuous at a?
5. State either definition of the derivative of f at x = a. Explain its meaning.
6. Name three distinct ways in which a function can fail to be differentiable at a.
7. Explain the difference between an absolute maximum and a local maximum.
8. What does the Extreme Value Theorem say?
9. What is the definition of a critical point?
10. What is the definition of an inflection point?
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1. (a) 2 (b) 2/3 (c) 1 (d) 1/5 (e) 0 (f) 0 (g) 1
2. The vertical asymptotes are x = 1 and x = −1. At x = 1, f approaches −∞ from the left
and ∞ from the right. At x = −1, f approaches ∞ from the left and −∞ from the right.
3. The horizontal asymptotes are y = 2 and y = −2. As x → ∞, f (x) approaches 2, and as
x → −∞, f (x) approaches −2.
1
1
4. Since −1 ≤ sin x ≤ 1, −1/x2 ≤ (sin x)/x2 ≤ 1/x2 . Furthermore, lim − 2 = lim 2 = 0, so
x→∞ x
x→∞ x
sin x
by the squeeze theorem, lim
= 0.
x→∞ x2
5. a = 1, b = 3
6. (a) x = 5, 7, 9 (b) x = 3, 5, 7, 8, 9
2x3 cos(x4 )
.
7. f 0 (x) = p
sin(x4 )
8. − sin(xy)(xy 0 + y) = y 0 cos(y).
9. y − 2 = 27 (x − 1).
10. Absolute minimum: -1, Absolute maximum: 27
11. Local min at x = −1 and x = 1, local max at x = 0. f is increasing
on (−1, 0) ∪ (1, ∞),
1
1
√
√
decreasing on (−∞, −1) ∪ (0, 1), concave up on −∞, − 3 ∪
, ∞ , and concave down
3
on − √13 ∪ √13 .
12. (a) f 0 is positive on (2, 6) and (8, 10) since f is increasing on those intervals. (b) f 00 is positive
on (0, 5) and (7, 9) because f is concave up on those intervals.
13. F (x) = 2 sin x + 3.
14. ex sec x
15. 3x2 sin(x6 )
Answers to Concept Questions
1. (a) f (x) + 2, (b) f (x + 3), (c) −f (x), (d) f (x/5), (e) f (2x)
2. arcsin x = y if and only if sin y = x. Its domain is −1 ≤ x ≤ 1 and its range is −π/2 ≤ y ≤
π/2.
3. If g(x) ≤ f (x) ≤ h(x) near x = a and lim g(x) = lim h(x) = L, then lim f (x) = L.
x→a
x→a
4. lim f (x) = f (a).
x→a
5
x→a
f (a + h) − f (a)
f (x) − f (a)
or f 0 (a) = lim
. The definition says that the slope
x→a
h
x−a
of the tangent line at x = a is the limit of the slopes of the secant lines connecting (a, f (a))
to the points approaching it.
5. f 0 (a) = lim
h→0
6. If the function is not continuous at a, if the function has a cusp/corner at a, or if the function
has a vertical tangent line at a. (Another acceptable answer would be “vertical asymptote,
vertical tangent line, cusp,” but I would not accept “vertical asymptote, discontinuity, cusp”
or “cusp, corner, discontinuity” for example, since a vertical asymptote is a type of discontinuity, and a cusp/corner is referring to the same type of issue.)
7. An absolute maximum is the maximum y-value that a function achieves. A local maximum is
a y-value on the function that is bigger than the ones near it (i.e. on an open interval around
it).
8. The Extreme Value Theorem says that a function which is continuous on a closed interval
achieves an absolute maximum and an absolute minimum on that interval.
9. A critical point of f is a value x such that f 0 (x) = 0 or f 0 (x) does not exist.
10. An inflection point of f is a point (x, f (x)) such that f changes concavity at x, i.e. f 00 (x)
changes sign.
True-False problems:
Chapter 2 Review: # 1, 3, 11, 19, 21
Chapter 3 Review: # 1, 3, 5, 7, 15
Chapter 4 Review: # 1, 3, 5, 7, 11
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