med-1K '01 72.37 .L SOL/Chews S'a Lia/7
1. Use the graph of the function at) shown at right to answer the following.
Part I: State the values of the following limits. If the limit is innite, please state whether
it is 00 or 00. If the limit does not exist
Chapter 1
Intro
Section 1.1: What is a
function?
It is like a black box
Input
The input of a function is called
Independent variable
We usually will use the letter x to
refer to the independent variable.
Sometimes we will use the letter t
but only if the
Math 1031 Midterm 2
Question 16
The derivative of f (x) = x + 4/x at x = 1 is 3. Find the best
approximation to f (1 h) when h is very small.
This problem is solved using . . . formula: . . .
Question 16
The derivative of f (x) = x + 4/x at x = 1 is 3. Fi
1
Chapter 1
Monotone functions
An increasing function preserves the order of inputs: f (x1 ) < f (x2 ), whenever x1 < x2 .
A decreasing function reverses the order of inputs: f (x1 ) > f (x2 ), whenever x1 < x2 .
Linear functions
Equation: f (x) = mx +
Math 1031 Midterm 2
1. This is the graph of f (x).
4
3
2
1
0
-1
-2
-1
0
1
2
One of the graphs below is the derivative of f (x). Which one is it?
1
10
0.5
5
0
0
-0.5
-5
-1
-10
-1.5
(A)
(C)
-2
-1
0
1
2
(B)
-15
-1
4
2
2
0
0
-2
-2
-4
-1
0
1
2
(D)
-1
0
0
1
1
2
Chapter 6
Antiderivatives
Section 7.1: Constructing
Antiderivatives Analytically
f
(
x
)
dx
F
(
x
)
d
F ( x) f ( x)
dx
Section 7.1: Constructing
Antiderivatives Analytically
Power rule
x p 1
C , p 1
k
p
k x dx p 1
k ln x C , p 1
Examples:
x,
5
3
, x ,
Chapter 3
Short-cuts to Differentiation
Section 3.1: Powers and
Polynomials
The derivative of the constant
function
d
d
0
0 1
k k x k 0 x 0
dx
dx
The derivative of a line is its
slope
d
mx b m
dx
Constant multiple, sum
and difference
If c is a constant
Chapter 4
Using The Derivative
Section 4.1: Local
Maxima and Minima
has a local
maximum
f ( p ) at p if
is greater than or
f (x)
equal to the values of
for points near
p f (x)
has a local
f ( p)
minimum at f p(x) if
is less than or
equal to the values of
Chapter 5
Accumulated Change: The Definite
Integral
Section 5.1: Distance and
Accumulated Change
How do we know the distance travelled?
Example: If you travelled at:
24 mph for 1 hour
33 mph for 1 hour
40 mph for 1 hour
45 mph for 1 hour
48 mph for 1
Chapter 2
Rate Of Change: The Derivative
Section 2.1:
Instantaneous Rate of
Recall: The Average Rate of Change
Change
The change in the value of a
quantity divided by the elapsed
time. For a function, this is the
change in the y-value divided by the
chang
Quiz 3 solutions.
Math 1041 Sections 015 & 019, Spring 2017
1. (6 pts) Find all horizontal asymptotes of the curve.
y=
(Hint:
x2 = |x|)
9x2 + 18x 3x
2x 5
The horizontal asymptotes can be found by finding the limits to infinity.
lim
x
q
9x2
+ 18x 3x
= lim
Math 1041 Sections 015 & 019, Spring 2017
Feb. 9th Notes for Snow day.
If you have questions regarding HW 2.6, 2.7, and 2.8, you can email me questions.
Just like how Limits have their Laws and Rules, Derivative also has a list of Rules.
The MOST IMPORT
Math 1041 Sections 015 & 019, Spring 2017
Feb. 16th Notes.
Reminders on Review Problems and the Review Session.
Q & A on 3.1, 3.2, and 3.3. Finding derivatives with given information, finding Tangent Lines of a
given slope, finding higher order derivati
Quiz 3.
2.6: Limits at Infinity & 2.72.8: Derivatives
Math 1041 Section
Spring 2017
Name:
Instruction: Justify your answers with work and/or explanation.
1. (6 pts) Find all horizontal asymptotes of the curve.
y=
2. (2 pts) Evaluate.
4
lim
x
9x2 + 18x 3x
Math 1041 Sections 015 & 019, Spring 2017
Practice problems for Quiz 4
Find the derivatives for the given functions:
1. f (x) = 3 +
f 0 (x) =
2. f (x) = x2
f 0 (x) =
3. f (x) = x5
x
f 0 (x) =
4. f (x) =
1
1
+
2
x
x
f 0 (x) =
5. f (x) = ex
f 0 (x) =
6.
Math 1041 Sections 015 & 019, Spring 2017
Practice problems for Quiz 4
Find the derivatives for the given functions:
1. f (x) = 3 +
f 0 (x) =
2. f (x) = x2
f 0 (x) =
3. f (x) = x5
x
f 0 (x) =
4. f (x) =
1
1
+
2
x
x
f 0 (x) =
5. f (x) = ex
f 0 (x) =
6.
MATH 1041 Review Problems for Test 1
2.2.4 Use the given graph (p.92) of f to state the value of each
quantity, if it exists. If it does not exist, explain why.
(a)
lim f (x) (b)
x2
2.3.49a Let g(x) =
x2 + x 6
. Find
|x 2|
2.3.49b Let g(x) =
x2 + x 6
. Do
Math 1041 Sections 015 & 019, Spring 2017
Feb. 14th Notes.
Reminders on Quiz 3, Quiz 4, WebAssign HW, Test 1 Sign-in sheet, and Review Problems for Test 1.
Where were we? The definition fo derivatives and the notation for derivatives.
Derivative Rules.
Quiz 2 and similar problems with solutions.
Math 1041 Sections 015 & 019, Spring 2017
1. The graphs of f and g are given. Use them to answer the questions below.
y
y
6
6
y = f (x)
y = g(x)
4
4
2
2
2
x
4
2
4
x
(a) (2 pts) Evaluate lim 4f (x) 3g(x)
x0
lim 4
Math 1041 Sections 015 & 019, Spring 2017
Jan. 24th Lecture Topics
The Laws for finding limits: (Suppose that f and g are functions with lim f (x) = L and lim g(x) = J.
xa
xa
Let c be a constant and n be a positive integer.)
i h
i
h
1. lim f (x) + g(x)
Math 1041 Sections 015 & 019, Spring 2017
Practice problems for Quiz 2
Evaluate the limit, if it exists. If it doesnt, consider whether the limit is .
lim +
x3
x2 + 3x
x2 x 12
Evaluate the limit, if it exists. If it doesnt, consider whether the limit is
Math 1041 Sections 015 & 019, Spring 2017
Jan. 26th Lecture Topics
Recap on 2.1 and 2.2. Homework Problems and Practice Quiz.
2.5: Continuity.
A function f is Continuous at x = a
if lim f (x) = lim+ f (x) = f (a).
xa
xa
To be specific, not only does the
Math 1041 Sections 015 & 019, Spring 2017
Feb. 7th Lecture Topics
Going over Practice Problems for 2.6 first. Infinite limits at infinity. Rationalize technique. Dealing
with square roots.
Test 1 on Feb 22nd, the materials should be
2.1: Introducing th
Math 1041 Sections 015 & 019, Spring 2017
Practice problems for Quiz 3
Find the Horizontal and Vertical Asymptotes of the curve.
y=
2x2 + 1
3x2 + 2x 1
Find the Horizontal and Vertical Asymptotes of the curve.
y = 4x2 + 3x 2x
Find the Horizontal and Ver
Math 1041 Sections 015 & 019, Spring 2017
Jan. 31st Lecture Topics
Recall Limits and Continuity with a piecewise function example with variables.
2
x 4
f (x) = a
bx
,x < 0
,x = 0
,x > 0
f is a continuous function over (, 0) because x2 4 is a polynomial.
Solutions to Quiz 1 and similar examples.
2.1 & 2.2
Math 1041 Section 019 Spring 2017
1. A ball is thrown into the air with a velocity of 20 ft/s, its height in feet t seconds later
is given by y = 5 + 20t 16t2 .
(a) (3 pts) Find the average velocity fo
Math 1041 Sections 015 & 019, Spring 2017
Practice problems for Quiz 1
The point P (2, 1) lies on the curve y = 1/(1 x). If Q is the point 1.9, 1/(1 1.9) , find the slope
of the secant line P Q.
If a ball is thrown into the air with a velocity of 40 ft/
Math 1041 Sections 015 & 019, Spring 2017
Jan. 19th Lecture Topics
Reminders: Email me to make an appointment for a short office visit. Review Quiz is now available
on WebAssign and start it when you are prepared.
Recall: lim f (x) the limit of f (x), a