# MATH 131:100 Exam 3 Review 27 June, 2014

```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:
Know all differentiation rules from Section 3.1.
Know Product, Quotient, and Chain Rules and how to compute derivatives using them.
Know Point-Slope Form equation for finding the equation of a line between two given
points.
Know derivatives of all basic functions.
Know derivatives of Trig functions (from section 3.3)
Know how to find the absolute maximum and minimum of a function on a closed interval.
Know how to find critical points of a function
Know first and second derivative tests
Understand how to formulate an optimization problem, and solve for the maximum and
maximum or minimum.
Know indefinite integrals of basic functions (or equivalently, the general antiderivatives)
Know how to solve for a function given its first or second derivative and information
Know how to compute a Left or Right rectanguler estimate for the area under a curve.
Know how to calculate definite integrals as the area under a curve
Know how to calculate definite integrals using the Evaluation Theorem (i.e. using the
antiderivative)
Know Comparison Properties of Integrals (at the end of Section 5.2)
Know integrals of all basic functions and trig functions
Know 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.
Know 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 .
```