Math 1310: CSM
Tutorial: Functions, continued
SOLUTIONS
Here we explore further features of Sage. To begin, please open up a new Sage worksheet
(call it whatever youd like; Functions 2, for example), OR you can use the worksheet that
you created in the pr
Math 1300 Section 1.4: Logarithmic Functions
1. Properties of Logarithms:
(a) Denition: logb (a) = x is exactly the same as bx = a for b > 1.
Notation: ln(x) = loge (x) and log(x) = log10 (x).
(b) The function logb (x) is the inverse of the function bx .
Math 1300 Section 1.3: New Functions From Old
1. Shifts, Stretches, Flips:
(a) If f (x) is a function, we can perform the following operations:
c f (x)
c f (x)
f (c x)
f (c x)
f (x) + a
f (x a)
f (x)
f (x)
vertical stretch by c if c > 1
vertical compressi
MATH 1300
Lecture Notes
Monday, August 26, 2013
1. Section 1.1 of HH - Functions and Change
(a) Functions: A function is a rule that relates two quantities, x (the independent
variable) and y (the dependent variable), such that each input (a value for x)
Math 1310 Quiz 8 Name: 3., luons
I1 1. (3.) Find / (28%.
gx 8X
3 2. Solve the initial value problem
More on the back -> Math 1310 Quiz 8 Name: SoiuHonS
3. Here is a graph of the function
Find the exact area of the starred region.
H
Area
{0 Pain
Math 1310 Quiz 5 Name: Soluons
d
Find Indicate any differentiation rules that you use. You do not need to simplify.
8x i la
1. 2 when ru
3 3008(1) - 53: + 2 1)
J )L] ___ 1 A .. +]
d A a* (3:055) - 5x + 2) 3;"): e (e )dJSoStx) 5" Z
: __ f g _
MATH 1300
Lecture Notes
Tuesday, August 27, 2013
1. Section 1.2 of HH - Exponential Functions
(a) Exponential Functions: An exponential function has the form y = abx , or using
function notation, f (x) = abx . Note that the variable, x, is in the exponent
Section 1.6 - Powers, Polynomials, and Rational Functions
Power Functions and Polynomials
Denition A power function has the form f (x) = kxp where k and p are constant.
Ex: Volume of a sphere, as a function of radius V = 4 r3
3
Note: Odd powers are seat s
MATH 1300
Lecture Notes
Section 2.6 - Dierentiability
Denition: A function f is called dierentiable at x if
f (x + h) f (x)
h0
h
lim
exists.
Note: Graphically, this means f has a well-dened, non-vertical tangent line (i.e., the
graph is smooth) at x.
W
Sec. 2.5: The Second Derivative
From previous sections, we know that if we have a dierentiable function f (x), we can
nd a derivative function f (x). But f (x) is a function itself (possibly dierentiable), which
raises the question what happens when we ta
MATH 1300
Lecture Notes
Wednesday, September 25, 2013
1. Section 3.1 of HH - Powers and Polynomials
In this section 3.1, you are given several dierentiation rules that, taken altogether,
allow you to quickly and easily dierentiate all power functions and
MATH 1300
Lecture Notes for Section 2.4
Friday, September 20, 2013
From the previous section, we learned that f (x) is the derivative function, which gives
the slope of the tangent line to the function at any given x-value. The purpose of this section
is
Math 1300: Calculus 1
Section 1.8: Limits
Sep 9-10, 2013
Limits
We say the limit of f (x) as x approaches c is L or
lim f (x) = L
xc
if f (x) can be made as close as we like to L by choosing x sufciently close to (but not equal to) c.
Note: f (c) itself n
1.7
Introduction to Continuity
The goal of this section is for us to determine if (and where) a function f (x) is continuous. To begin, here
is an informal denition of continuity:
Denition 1. A function f (x) is continuous if its graph can be drawn withou
MATH 1300
Lecture Notes
Monday, September 16, 2013
1. Section 2.3 of HH - The Derivative Function
The major dierence between this section 2.3 (The Derivative Function) and the previous section 2.2 (The Derivative at a Point) is that instead of computing t
SECTION 1.5: TRIGONOMETRIC FUNCTIONS
The Unit Circle
The unit circle is the set of all points in the xy-plane for which x2 + y 2 = 1.
Def: A radian is a unit for measuring angles other than degrees and is measured
by the arc length it cuts o from the unit
Solutions to Selected Exercises: Individual Homework Assignment #1
Assignment: pp. 19-27, exercises 15, 16, 17, 19, 21.
Exercise 15
We consider once again the specic rate equations:
S0 =
0.00001S I
I 0 = 0.00001S I
1
I
14
1
I
14
As discussed in the text o
Math 1310
Exam 2 Review 2
Name:
1. Find dy/dx. Indicate any dierentiation rules that you use.
(a) y = tan sin 4x3
dy
d
=
tan sin 4x3
dx
dx
d
chain rule
sin 4x3
dx
d
cos 4x3
chain rule again
4x3
dx
cos 4x3 12x2
= sec2 sin 4x3
= sec2 sin 4x3
= sec2 sin 4
MATH 1310: CSM
EXAM 2 REVIEW
1. Fill in the dashed rectangles (there are six of them) on the picture below, by inserting exactly
one of the following terms in each rectangle.
y
x
slope
a
secant
tangent
negative
f (a)
Note: there are more terms than there
MATH 1310: CSM
February 5, 2014
EXAM 1: SOLUTIONS
1. On Monday February 2 (Groundhog Day!), there were 16 inches of snow on the ground. By
that Saturday (February 7), there were only 6 inches remaining. Use this information to
answer the following questio
Math 1310: CSM
Sample exam 1 problems
SOLUTIONS
PLEASE NOTE that the rst exam will cover all material from Sections 1.1 and 1.2, and the
corresponding tutorials.
1. Let
1
, g (t) = t3 ,
x2
Find the following function values:
f (x) =
p(z ) = z 2
3z ,
q (v
Math 1310: CSM
Sample exam 1 problems
SOLUTIONS
PLEASE NOTE that the rst exam will cover all material from Sections 1.1 and 1.2, and the
corresponding tutorials.
1. Let
f (x) =
x2
4
x
,
g (x) = 2x,
and
h(x) = 3
x.
Find the following values, and express in
Math 1310
Basic Derivative Rules & Chain Rule
In each problem, nd a formula for
dy
. Do not use the chain rule.
dx
1. y = 3x + 8
dy
=3
dx
2. y = 100
dy
=0
dx
3. y = x30
dy
30
= 30x31 = 31
dx
x
4. y = 5x
dy
= k5 5x
dx
5. y = 7x2 + 7x + 15x4
dy
= 14x + k7 7