Math 180: Calculus I
Fall 2014
September 9
TA: Brian Powers
Facts we can use
lim
x
1
1
= 0, lim
= 0.
x
x
x
By limit laws we then have
c
=0
xn
for any constant c and for any positive exponent n (whether it is an integer or a fraction). This is very
handy!
Math 180: Calculus I
Fall 2014
October 28
TA: Brian Powers
When graphing functions, there is some basic analysis that will help you do it.
Identify the domain or the interval in question
identify if there is any helpful symmetry (e.g. even/odd function)
Math 180: Calculus I
Fall 2014
November 25
TA: Brian Powers
1. Use symmetry to evaluate these integrals
R /4
(a) /4 cos xdx
SOLUTION: Since the cosine function is an evan function, and the interval is syummetric to
the y-axis, it suffices to double the in
Math 180: Calculus I
Fall 2014
October 16
TA: Brian Powers
1. Sketch a function that is continuous on (, ) with the following conditions: f 0 (1) is undefined;
f 0 (x) > 0 on (, 1); f 0 (x) < 0 on (1, ).
SOLUTION: Though there are many possible graphs, he
Math 180: Calculus I
Fall 2014
October 9
TA: Brian Powers
We may use the following derivative rules now:
d
1
sin1 x =
dx
1 x2
d
1
tan1 x =
dx
1 + x2
d
1
sec1 x =
dx
|x| x2 1
d
1
, for 1 x 1
cos1 x =
dx
1 x2
d
1
cot1 x =
dx
1 + x2
d
1
, for |x| > 1
csc1
Math 180 Week 8 Tuesday
1. Compute the derivative of the following functions.
(a) y = (tan-1x)?
y'= italm - -L
('13
Fall 2015
2. A baseball diamond is in the shape of square. Each side has a length of 90 feet. A player is
running from rst to second base a
1 Computing limits
1. Below are six graphs.
0/ 0/ ./ ./ / /
/ /' /O /
l 2 3 4 5 6
Which of these graphs satisfy the properties below?
(a) the graph is a function
1,2,3,5,6
(b) the graph is dened at every point
2,3,4,6
(c) lefthand-side limits exist at eve
MTH 201
Chapter 6: Application of Integration
Spring 2016
6.7 Moments and Centers of Gravity
General Idea
Suppose we have a flat plate (called a lamina) of some material with uniform density which occupies a region
of the -plane. A picture of a possible p
MTH 201
Chapter 6: Application of Integration
Spring 2016
6.1 Areas Between Curves Worksheet
1. Find the area of the region enclosed by the curves = 2 and = 2 2 .
Find Intersections:
2 = 2 2
Setup:
2 2 2 = 0
2( 1) = 0
= 1 = 0
(1,1)
1
= [(2 2 ) 2 ]
0
E
Math 180: Calculus I
Fall 2014
October 14
TA: Brian Powers
1. On the following graph to determine at what x values on the interval [a, b] local and absolute extreme
values occur.
SOLUTION: Local minima at x = q. Local maxima at x = p, r. Absolute minimum
Math 180: Calculus I
Fall 2014
September 11
TA: Brian Powers
Example 4.1 Is the following function continuous at a?
1. f (x) =
2x2 +3x+1
x2 +5x ; a
2. f (x) =
x2 1
x1
3
=5
if x 6= 1
;a = 1
if x = 1
1. This is a rational function, and rational functions ar
Math 180: Calculus I
Fall 2014
September 2
TA: Brian Powers
1.1
Limits of Special Functions
1. Constant Functions: if f (x) = c, then limxa f (x) = c
2. Linear, Polynomial, Sine, Cosine: limxa f (x) = f (a)
3. Rational Functions: Factor, Cancel, Hope (tha
1
Properties of limits
1. Let f, g be two functions with lim f (x) = 2 and lim g(x) = 6. Showing all your steps,
xc
xc
simplify the following limits.
(a) lim[8g(x)]
xc
lim[8g(x)] = 8 lim g(x) = 8 6 = 48
xc
xc
(b) lim[5f (x) + 9g(x)]
xc
lim[5f (x) + 9g(x)]
1
Functions and slopes
1. Consider the following conditions on a function f :
the domain of f is all real numbers
f (x) 0 for any real number x
f (4) = f (4)
Answer the questions below using different functions in each part.
(a) Give an example of a fu
Math 180: Calculus I
Fall 2014
September 18
TA: Brian Powers
1. Find the derivatives of the following functions.
(a) f (x) = 6x 2xex
SOLUTION: Use the sum/difference rule, followed by a product rule
d
d
6x |cfw_z
2x |cfw_z
ex = 6 (u0 v + v 0 u) = 6 (2)ex
Math 180: Calculus I
Fall 2014
September 23
TA: Brian Powers
1. Evaluate the following limits using the facts that:
lim
x0
(a) limx0
sin x
=1
x
and
lim
x0
cos x 1
=0
x
tan 5x
x
(b) limx3
sin(x+3)
x2 +8x+15
2. Evaluate the derivatives dy/dx
(a) y =
cos x
s
Math 180: Calculus I
Fall 2014
November 4
TA: Brian Powers
1. Use linear approximation to first find the derivative at x = a, then estimate f at the given point.
(a) f (x) = 12 x2 ; a = 2, f (2.1)
SOLUTION: f 0 (x) = 2x, and at x = a, f (2) = 12 4 = 8, f
Math 180: Calculus I
Fall 2014
November 18
TA: Brian Powers
1. Express P
as a definite integral:
n
lim0 k=1 (x2
k + 1)xk on [0, 2]
R4
R6
2. Suppose 1 f (x)dx = 8 and 1 f (x)dx = 5.
Evaluate the following integrals
(a)
R4
(b)
R4
1
6
(3f (x)dx
5. Remember t
Math 180: Calculus I
Fall 2014
October 30
TA: Brian Powers
1. Two nonnegative numbers x and y have a sume of 23. What is the maximum possible product?
2. A boxs total dimensions (length + width + height) cannot exceed 108 in. If the box has a square
base,
Math 180: Calculus I
Fall 2014
October 21
TA: Brian Powers
At this point you should have the following derivatives memorized.
f (x)
g(x) + h(x)
g(x) h(x)
g(x)h(x)
g(x)/h(x)
g(h(x)
c
xn
ex
bx
ln x
logb x
sin x
cos x
tan x
sec x
cot x
csc x
sin1 x
cos1 x
ta
Math 180: Calculus I
Fall 2014
October 30
TA: Brian Powers
1. Two nonnegative numbers x and y have a sume of 23. What is the maximum possible product?
SOLUTION: We wish to maximize p = xy, but first we would like to express this as a function of a
single
MTH 201
Chapter 6: Application of Integration
Spring 2016
6.6 Work
The word work is often interpreted to mean the amount of effort required to perform a task.
When used in the area of physics, the term work is an idea that is based off the concept of forc
MTH 201
Chapter 6: Application of Integration
Spring 2016
6.2 Volumes of Revolutions (Disk & Washer Method)
General Idea:
In this section we are interested in trying to use our knowledge about integral and calculating areas to help
compute the volumes of
1. Order the slopes of the tangent lines to the graph at the indicated points:
f 0(
) < f 0(
) < f 0(
) < f 0(
)
2. Answer the following question at the indicated points on the graph:
(a) At which points if f continuous?
1
(b) At which points is f dierent
Math 221/AL1 Exam II
UIUC, October 31, 2013
1. (8 points) A cold drink is taken from a refrigerator, its temperature is 5 C.
After 25 minutes in a 20 room, it temperature has increased to 10 . What is
the temperature of the drink after 35 minutes?
2. (10