Review for Math 115 midterm 1
Here is a brief list of topics you should review for the first exam (Note: anything we have had
homework on or done in class is fair game, even if it is not expressly described below. The list
below is merely intended to help you organize your studying and to remind you of some of the most
important topics.). The midterm covers sections 4.9, 5.1-5.5, and 6.1.
• Derivatives you should know:
d n
d x
[x ] = nxn−1
[e ] = ex
dx
dx
d
[sin x] = cos x
dx
d
[cos x] = − sin x
dx
d
[tan x] = sec2 x
dx
d
[sec x] = sec x tan x
dx
d
[csc x = − csc x cot x]
dx
d
[cot x] = − csc2 x
dx
d
1
[ln x] =
dx
x
d
1
[arctan(x)] =
dx
1 + x2
You should also know the product rule, the quotient rule, and the chain rule.
• Antiderivatives/Indefinite Integrals
You should know the following antiderivatives:
Z
Z
1
xn+1
n
+ C, (n 6= 1)
dx = ln |x| + C
x dx =
n+1
x
Z
Z
ex dx = ex + C
sin x dx = − cos x + C
Z
Z
cos x dx = sin x + C
Z
Z
tan x dx = ln | sec x| + C
sec x tan x dx = sec x + C
Z
2
csc x dx = − cot x + C
Z
sec2 x dx = tan x + C
1
dx = tan−1 x + C
1 + x2
• Riemann sums–estimating
Z
csc x cot x dx = − csc x + C
– You should know how to estimate definite integrals/signed areas by using Riemann sums
with right or left endpoints. You should be able to sketch the graph and the associated
rectangles. You should also know what the notation Rn and Ln mean. You should be
able to compute Riemann sums for integrals where the integrand is defined by a formula,
given by a graph, or given by a table of values.
• Problems involving position, velocity, and acceleration
• Problems where you need to find f (x) if you are given f 0 (x) or f 00 (x) and you know the value
of f (a) and/or f 0 (a) for some number a.
• Definite integrals and what they represent graphically
– How to use a picture to evaluate a definite integral
• How to evaluate definite integrals in general
– You will need to determine what method to use (drawing a picture, using the Fundamental Theorem of Calculus, Using a u-substitution).
• Properties of definite integrals
Z b
Z a
f (x) dx = −
f (x) dx
*
a
b
Z
b
*
Z
f (x) dx +
a
Z
c
Z
f (x) dx =
b
b
*
Z
*
f (x) dx
a
Z
cf (x) dx = c
a
c
b
f (x) dx
a
b
Z
[f (x) + g(x)] dx =
a
b
Z
f (x) dx +
a
b
g(x) dx
a
• How to use the Fundamental Theorem of Calculus Part I
– Don’t forget to practice problems that use the Chain Rule or the Properties of Definite
Integrals
• U-substitutions
– Make sure that you know how to use u-substitution for both indefinite and definite
integrals. For definite integrals, don’t forget to change the limits of integration from x
labels to u-labels.
Z b
0
• Net Change Theorem: If f (x) is a rate of change, then
f 0 (x) dx gives the net change in
a
f between x = a and x = b. In words: integrating a rate of change from a to b gives the
amount of change from a to b.
– Know how to use the Net Change Theorem to explain what expressions or equations
involving integrals represent.
– Know how to use the Net Change Theorem to find how much something has changed or
the current value of a function.
– Know how to find the displacement and total distance travelled by an object if you know
the velocity function (don’t forget that total distance involves integrating an absolute
value function - make sure you know how to do that!)
• Average Value of a function: The average value of f (x) on the interval [a, b] is given by
Z b
1
f (x) dx.
b−a a
– Make sure that you know how to find the average value of a function on an interval.
Make sure that you know the formula.
Note: Any problem that is similar to examples done in class or problems in the homework (suggested
or Webwork) is fair game. There may also be a problem that is not totally similar to anything we
have done but instead tests your understanding of the concepts.
To study: Look at the examples done in class. Go over your Webwork. Test your understanding
of the material by doing the practice problems below. For any section that you are still having
trouble with, do suggested problems.
1. Evaluate
Z 1
p
(a)
(x + 1 − x2 ) dx [Hint: split into 2 integrals, and use a picture for the second one.]
Z0
(b)
sec x(sec x + tan x) dx
Z
x+2
√
(c)
dx
x2 + 4x
Z arctan x
e
(d)
dx
1 + x2
Z
(e)
sec(3x) tan(3x) dx
Z
ln(8)
(f)
√
ex 1 + ex dx
ln(3)
Z
(g)
Z
(h)
0
Z
(i)
x5
p
5
x3 + 1 dx
2
(x2 − |x − 1|) dx
√
e x
√ dx
x
Z
6
f (x) dx
2. Use the graph of f to calculate R3 and L3 approximations of
0
3. Evaluate
Z 1
d arctan x (a)
e
dx
0 dx
Z 1
d
arctan x
e
dx
(b)
dx 0
Z x
d
arctan t
(c)
e
dt
dx 0
Z
5
4. Differentiate the function g(x) =
p
t2 + sec(t) dt
sin x
5. Let r(t) be the rate at which the world’s oil is consumed where t is measured in years starting
Z 8
at t = 0 on January 1, 2000, and r(t) is measured in barrels per year. What does
r(t) dt
0
represent?
Z
4
6. If f (1) = 12 and
f 0 (x) dx = 17, what is f (4)?
1
Z
7. Let g(x) =
x
f (t) dt where g is the function whose graph is below.
0
(a) At what values of x do the local maximum and minimum values of g occur?
(b) What is g 0 (5)?
(c) On what intervals is g concave downward?
2
8. Find the average value of te−t on the interval [0, 5]
9. Use the graph from problem 2: What is the average value of f 0 (x) for 2 ≤ x ≤ 4?
Answers:
1 π
+
2 4
(b) tan x + sec x + C
√
(c) x2 + 4x + C
1. (a)
(d) earctan x + C
(e)
1
3
sec(3x) + C
38
(f)
3
5 3
1 5 3
11/5
6/5
(x + 1)
− (x + 1)
+C
(g)
3 11
6
(h) 5/3
√
(i) 2e
x
+C
2. R3 = 2(4) + 2(−1) + 2(−3)
L3 = 2(2) + 2(4) + 2(−1)
3. (a) earctan(1) − earctan(0)
(b) 0
(c) earctan x
p
4. g 0 (x) = − sin2 x + sec(sin x)(cos x)
5. The number of barrels of oil consumed from January 1, 2000 through January 1, 2008.
6. 29
7. (a) local max at x = 1 and x = 5; local min at x = 3 and x = 7
(b) 0
(c) ( 12 , 2), (4, 6), (8, 9)
8.
1
(1 − e−25 )
10
9. −
5
2