Math 221-Final Exam Review Guide
Ruifang Song
University of Wisconsin-Madison
May 3, 2012
1. Old material
Please refer to Midterm 1 & 2 review guides for a review of old material. You are
also encouraged to go over our two midterms. Note: Linear approximation,
Newton’s method and related rates will NOT be on the final.
2. Area and estimating with finite sums
• To be able to estimate areas by using rectangles.
• To understand the notions of right/left/midpoint sum estimate.
• To be able to find the exact area (in simple cases) by computing it as the
n
n
X
n(n + 1) X 2
k =
,
limit of Riemann sums. Note: The formulas:
k =
2
k=1
k=1
n
n(n + 1)(2n + 1) X 3
n(n + 1) 2
k =(
,
) , if needed, will be given on the final
6
2
k=1
exam.
3. Definite integrals
• To understand the notion of a Riemann sum.
• To understand the definition of a definite integral as the limit of Riemann
sums when the norm of the partition approaches 0.
Z b
• To understand that the definite integral
f (x) dx gives the signed area (we
a
consider areas below the x-axis as negative) bounded between the graph of
f (x) and the x-axis over [a, b].
• To be familiar with properties of definite integrals as in Table 5.4 on page
266 of the textbook, in particular, the max-min inequality and domination
property.
1
• To be able to evaluate a definite integral
special cases) by computing the
Z (in
1 √
π
corresponding area geometrically. e.g.
1 − x2 dx = as it is the area
2
−1
of the top half of the unit circle.
• To be able to use symmetry
Z definite integrals: If f (x)
Z when evaluating certain
4
a
sin(x3 ) dx = 0
f (x) dx = 0. E.g.
is an odd function, then
−4
−a
1
• To understand that the average value of a function is
b−a
Z
b
f (x) dx
a
• To understand the Mean Value Theorem for definite integrals:
If f (x) is continuous over [a, b], then there exists a point c in [a, b] such that
Z b
1
f (x) dx = f (c).
b−a a
4. The Fundamental Theorem of Calculus (FTOC)
• To understand Part
Z xI of FTOC: If f (x) is continuous on [a, b], then the area
function A(x) =
f (t) dt is differentiable over [a, b]. Moreover, A0 (x) =
a
f (x).
• To be able to use the above and
the chain rule to compute derivatives of
Z sin x
Z √x
functions like
f (t) dt or
f (t) dt
a
sin x
• To understand and be able to use Part II of FTOC(evaluation theorem):
If f (x) is continuous on [a, b] and F (x) is an antiderivative of f (x), then
Z b
f (x) dx = F (b) − F (a).
a
5. Substitution for indefinite integrals and definite integrals
• To understand the substitution method as a reverse to the chain rule. To be
able to apply substitution for indefinite integrals:
Z
Z
0
f (g(x))g (x) dx = f (u) du,
where u = g(x) and du = g 0 (x) dx.
• To be able to apply substitution for definite integrals:
Z
b
0
Z
g(b)
f (g(x))g (x) dx =
a
f (u) du,
g(a)
2
where u = g(x) and du = g 0 (x) dx.
Caution: Don’t forget to change the bounds of the integral accordingly. Or equivalently, you can first find the indefinite integral, then
re-substitute, then evaluate.
Z 2
2
2xex dx, let u = x2 . Then du = 2x dx.
Example 1 To compute
0
Z
Method 1:
2
2xex dx =
Z
Z
2
eu du = eu + C = ex + C. Thus
2
2
2
2xex dx = ex |20 = e4 − 1.
0
Z
2
2xe
Method 2:
x2
Z
dx =
0
4
eu du = eu |40 = e4 − e0 = e4 − 1.
0
6. Area between curves
To be able to compute the area bounded by two or more curves.
• First sketch the region and find out where different curves intersect each other.
• Represent the area of designated region as an integral (or a sum of integrals)
in terms of x (or in terms of y). Sometimes it is easier to do it one way than
the other.
• Evaluate.
√
Example 2 To find the area bounded by y =Z x and y = x − 2 and the x-axis, it
2
(y + 2 − y 2 ) dy. If you write it in
is easier to represent the area as an integral
0Z
Z 4
2√
√
terms of x, it will be the sum of two integrals
x dx +
[ x − (x − 2)] dx.
0
2
Example 3 For the area bounded between the graph of y = sin−1 (x) and the x-axis
Z π/2
over [0, 1], integrating in terms of y gives you
(1 − sin y) dy, while integrating
0
Z 1
in terms of x gives you
sin−1 x dx. We like the first approach because the
0
integral in terms of y is easier to evaluate.
7. Computing volumes by using cross-section area
To understand the idea of slicing and be able to apply the cross-section formula.
For a solid which lies between
x = a and x = b, if its cross-section area at x is
Z
b
A(x), then its volume is
A(x) dx.
a
3
8. Volumes of revolution
To be able to find the volume of a solid obtained by rotating a region about a
horizontal line or vertical line. The strategy is as follows. (Please follow this
strategy, instead of just memorizing formulas). For more information, see the
Volumes of revolution handout.
• Slice the region into n slices of equal width (either horizontally or vertically,
depending on which way is easier).
• Focus on the k-th slice. Estimate the volume that results from rotating the
k-th slice.
• Sum up over all slices to get a Riemann sum which gives an estimate for the
total volume of the solid.
• Take the limit as n goes to infinity and express the limit as a definite integral.
• Evaluate the definite integral.
9. Inverse functions and their derivatives
• To understand the definition of inverse functions. Inverse functions are defined only for one-on-one functions or one-on-one pieces of functions. e.g. you
can not define the inverse of y = x2 over (−∞, ∞), but you can define its
inverse over [0, ∞).
• To be able to find the inverse of a given function f (x). To understand that
their graphs are symmetric with respect to the line y = x.
• To understand that f (f −1 (x)) = x and f −1 (f (x)) = x.
• To distinguish the inverse function f −1 (x) from [f (x)]−1 =
1
.
f (x)
• To understand and be able to apply the formula for computing derivatives of
d −1
1
inverse functions:
f (x) = 0 −1
.
dx
f (f (x))
• To be familiar with the inverse trig functions sin−1 x, cos−1 x, tan−1 (x). To
know their values at special points and their derivatives.
10. ln x and ex
Z
• To understand the definition of ln x as
0
x
1
dt.
t
• To be familiar with the graph of ln x. In particular, ln 1 = 0, ln e = 1,
lim ln x = −∞, lim ln x = ∞.
x→0+
x→∞
• To understand and be able to apply algebraic properties of ln x.
4
• To understand and be able to apply logarithmic differentiation.
• To understand that ex and ln x are inverse to each other.
• To be familiar with the graph of ex . In particular, e0 = 1, lim ex = 0,
x→−∞
lim ex = ∞.
x→∞
• To understand and be able to apply algebraic properties of ex .
• To understand general exponential functions ax (= ex ln a ) and be able to find
their derivatives.
11. L’Hopital’s rule
In what follows, x → a can be replaced by x → a+, x → a−, x → ∞, x → −∞.
Caution: Before you apply L’Hopital’s rule, you need to verify that you
have an indeterminate form. Otherwise, you would get the wrong limit!
•
0
indeterminate form: Suppose f, g are differentiable on an open interval
0
around a and f (a) = g(a) = 0, then
f (x)
f 0 (x)
= lim 0
,
x→a g(x)
x→a g (x)
lim
assuming the right hand side limit exists or is ∞ or −∞.
∞
indeterminate form: Suppose f, g are differentiable on an open interval
•
∞
near a, lim f (x) = ∞ or −∞, and lim g(x) = ∞ or −∞, then
x→a
x→a
f (x)
f 0 (x)
= lim 0
,
x→a g(x)
x→a g (x)
lim
assuming the right hand side limit exists or is ∞ or −∞.
• 0·∞ indeterminate form: Rewrite the product as a quotient so that it becomes
0
∞
indeterminate form.
either indeterminate form or
0
∞
• 1∞ , 00 , ∞0 indeterminate form: By taking natural log of the original expression, transform it into one of the above three forms.
0
• ∞ − ∞ indeterminate form: Rewrite the expression in terms of either
0
∞
indeterminate form or
indeterminate form.
∞
• Summary of methods of computing limits: by using limit laws and direct
substitution property, by using algebraic manipulations, Sandwich theorem
(see midterm 1 review guide); L’Hopital’s rule (applies only to indeterminate
forms)
5
12. Derivatives and antiderivatives you need to know
d n
d
d
d
x = nxn−1 ;
sin x = cos x,
cos x = − sin x,
tan x = sec2 x;
dx
dx
dx
dx
d
1 d x
d x
•
ln |x| = ,
e = ex ,
a = ax ln a, where a > 0.
dx
x dx
dx
d
1
d
1
d
1
•
sin−1 x = √
,
cos−1 x = − √
,
tan−1 x =
.
2
2
dx
1 + x2
1 − x dx
1 − x dx
•
The above list of derivatives also gives you a list of antiderivatives:
Z
Z
1
xn+1
n
+ C, where n 6= −1,
dx = ln |x| + C (don’t forget
•
x dx =
n+1
x
the absolute value!);
Z
Z
Z
•
cos x dx = sin x + C, sin x dx = − cos x + C, sec2 x dx = tan x + C;
ax
+ C;
ln a
Z
Z
1
1
−1
√
•
dx = tan−1 x + C.
dx = sin x + C,
1 + x2
1 − x2
Z
•
x
x
e dx = e + C,
Z
ax dx =
6