MATH 131:100 Exam 3 Review
27 June, 2014
This Exam will have three sections: one of Definitions, one for graphical concepts, and
a section of workout problems. For this Review, I will just give a sample of the types of
problems that I could ask, but any of the material we have covered From Section 4.2 to
Section 5.5 is fair game.
Definitions You Should Know:
1) The Fundamental Theorem of Calculus:
Suppose f isZcontinuous on [a, b]. Then
x
a) If g(x) =
f (t)dt, then g 0 (x) = f (x)
a
Z b
b)
f (x)dx = F (b) − F (a), where F is any antiderivative of f , that is, F 0 = f .
a
2) An Antiderivative of a function f is a function F (x) such that F 0 (x) = f (x) for all x
in the domain of f .
3) The Mean Value Theorem:
If f is a differentiable function on the interval [a, b], then there exists a number c between
a and b such that
f (b) − f (a)
f 0 (c) =
b−a
4) A number c is a Critical Point of a function f if either f 0 (c) = 0 or f 0 (c) is undefined.
5) We say that f (c) is a local maximum of f if f (c) ≥ f (x) when x is near c.
6) We say that f (c) is a local minimum of f if f (c) ≤ f (x) when x is near c.
7) We say that f (c) is an absolute maximum of f if f (c) ≥ f (x) for all x in the domain
of f .
8) We say that f (c) is an absolute minimum of f if f (c) ≤ f (x) for all x in the domain
of f .
Other Topics:
ˆKnow all differentiation rules from Section 3.1.
ˆKnow Product, Quotient, and Chain Rules and how to compute derivatives using them.
ˆKnow Point-Slope Form equation for finding the equation of a line between two given
points.
ˆKnow derivatives of all basic functions.
ˆKnow derivatives of Trig functions (from section 3.3)
ˆKnow how to find the absolute maximum and minimum of a function on a closed interval.
ˆKnow how to find critical points of a function
ˆKnow first and second derivative tests
ˆUnderstand how to formulate an optimization problem, and solve for the maximum and
minimum asked for. (Section 4.6) This includes showing that your answer is indeed the
maximum or minimum.
ˆKnow indefinite integrals of basic functions (or equivalently, the general antiderivatives)
ˆKnow how to solve for a function given its first or second derivative and information
about the function itself.
ˆKnow how to compute a Left or Right rectanguler estimate for the area under a curve.
ˆKnow how to calculate definite integrals as the area under a curve
ˆKnow how to calculate definite integrals using the Evaluation Theorem (i.e. using the
antiderivative)
ˆKnow Comparison Properties of Integrals (at the end of Section 5.2)
ˆKnow integrals of all basic functions and trig functions
ˆKnow how to use the Fundamental Theorem of Calculus to differentiate integrals. Also
know how to do this when the limits are functions of x rather than x itself.
ˆKnow how to use the Substitution Method to evaluate integrals.
Example Problems:
Problem 1) Let
f 00 (x) = 3ex + x−2
and suppose that f 0 (0) = 5 and f (0) = 3. Find f (x).
Problem 2) Evaluate
Z 1
3
x + √ + x + −6x + 7 dx
x
5
2
Problem 3) Evaluate
Z
ex
dx
ex + 1
Problem 4) Evaluate
Z
4
1
√
e x
√ dx
x
Problem 5) Evaluate
1
Z
(3t − 1)50 dt
0
Problem 6) Evaluate
Z
π/2
−π/2
x2 sin x
dx
1 + x6
Problem 7) The sum of two positive numbers is 16. What is the smallest possible value
of the sum of their squares?
Problem 8) A model used for the yield Y of an agricultural crop as a function of the
Nitrogen level N in the soil (measured in the appropriate units) is
Y =
kN
1 + N2
where k is a positive constant. What Nitrogen level gives the best yield? What is the yield?
Problem 9) If
Z
ex
f (x) =
sin xdx
x2
then find f 0 (x).
Problem 10) Find the absolute maximum and minimum of the function y = (x2 − 1)3
on the interval [−1, 2].
Problem 11) If
f (x) = 2x3 + 3x2 − 36x
a) Find the intervals on which f is increasing or decreasing.
b) Find the local maximum and minimum values of f .
c) Find the intervals of concavity and the inflection points of f .