Math 150A, Jagodina
Chapter 3
Tutors May Help
The Derivative
1
Introduction to the Derivative
Consider a function f and a line that passes through the points
c, f (c) and c + x, f (c + x) .
Denition: The average rate of change in f on the interval [c, c +

9.3 Maxima and Minima of Functions of Two Variables
If a function z f ( x , y ) has a relative maximum or a relative minimum at a critical point ( a , b ) , then both
f x ( a , b) 0 and f y (a , b) 0 .
If f ( x , y ) has any relative extrema, they will oc

4.3 The Chain Rule
If y f g ( x) , then y f g ( x) g ( x)
In words, the chain rule tells us to first take the derivative of the outer function, and then multiply it by the
derivative of the inner function.
Chain Rule :
Find the derivative of the flowing f

3.3 & 3.4 Rates of Change and the Derivative
One major area of calculus is concerned with how quantities are changing in what direction and how fast. For
example, if a company raises the price of a product, what happens to the revenue; or how profit chang

4.4 Derivatives of Exponential Functions
Basic Rules for Exponential Functions
If f ( x) e x , then f ( x) e x .
If f ( x) a x , then f ( x) ln a a x .
Differentiate the following functions.
Ex 1: f ( x) 12 e x
y x3 e x
y 5x
w 3 t 4t 2
Ex 2:
y x3 e x
P( x

3.1 Limits
Limits at finite points
Consider a function f (x) and a number a that may or may not be in the domain of f (x). Limits are used to
describe the behavior of f (x) near the number a .
Ex1: Let f ( x) 3 x 1 . If x gets very close to 2, what is f(x

4.1 Techniques for Finding Derivatives
There are several algebraic techniques (i.e., rules) for finding derivatives that can be used instead of the
f ( x h) f ( x )
definition of the derivative. These techniques give the same results as f ( x) lim
.
h0
h

4.5 Derivatives of Logarithmic Functions
Basic Rules for Logarithmic Functions
If f ( x) ln x , then f ( x )
1
x
If f ( x) log a x , then f ( x)
Differentiate the following functions.
Ex 1: f ( x ) 15 ln x
1
.
(ln a ) x
y x e 2 x 5ln x
y 3log 4 x
5
w(t

5.1: Increasing and Decreasing Functions
A function f is increasing
if f ( x1 ) f ( x2 ) whenever x1 x2 , i.e., the function values
as x
.
A function f is decreasing
if f ( x1 ) f ( x2 ) whenever x1 x2 , i.e., the function values
as x
.
f
on:
f
on:
Now, d

ALGEBRA SYLLABUS AND LECTURE TIMETABLEThe algebra course for MATH1151
is based on the MATH1151 Algebra Notes, which are essentialreading and must be
brought to all algebra tutorials. There is very little overlap between thissyllabus and
the algebra speciF

Chapter 9: Multivariable Calculus
9.1: Functions of Several Variables
A multivariable function describes a dependent quantity on two or more independent variables. For example,
production levels depend on two independent variables of capital and labor. Ma

8.3 Continuous Money Flow
Suppose that the continuous function f (t) describes a rate of flow of money, in dollars per year. This could be
money that youre depositing in a savings account or investment, or money thats being spent in annual
payments.
We le

7.5 The Area between Two Curves
Case 1: Given two functions and two vertical lines.
y
If f ( x) and g ( x) are continuous
functions with f ( x) g ( x) on [a, b ] ,
then the area between the curves from
x a to x b is given by
y f ( x)
y g ( x)
b
f ( x) g

Math 150A, Jagodina Sample Final Exam (1.1—5.7)
Name: Date: Version B
Show work for credit when applicable. Use methods similar to those demonstrated in class.
Answers without adequate justiﬁcation will not receive full credit. All solutions must be in
si

Math 150A, Jagodina
Chapter 4
Tutors May Help
Applications of the Derivative
1
Extreme Values of Functions
Denition:
Let f be dened on an open interval I containing c.
1. f (c) is the minimum of f on I if f (c) f (x) for all x I.
2. f (c) is the maximum o

Chapter 2
Section 2.1: A Preview of Calculus
The Tangent Line Problem: Differentiation
Secant lines approximate
the slope of the line
tangent to a curve at a
particular point. As we
move closer and closer to
the point, we get a closer
approximation to the

Appendix C
Section C.3: Review of Trigonometric Functions
Fill in the unit circle:
Degree to Radian Conversion: radians = 180
Ex] Rewrite
5
using degrees.
4
Ex] Rewrite 215 using radians.
1
Ex] Evaluate the sine, cosine, and tangent without using a calcul

Chapter 5
Section 5.1: Antiderivatives and Indefinite Integration
Definition:
antiderivative: a function F is an antiderivative of f on an interval I if F ' x f x for all x in I .
2
Question: What is the derivative of f x 3x 2 x 7 ?
Answer:
Question: What

6.1 Absolute Extrema (largest / smallest values)
Definition: Let f be a function defined on some interval. Let c be a number in the interval. Then f (c) is the
absolute maximum of f on the interval if f ( x) f (c) for every x in the interval.
Or f (c) is

5.2:
Relative Extrema (local Maximum or local Minimum)
Relative extrema (extrema is the plural of extremum) are the local highest points and the local lowest
points on a graph or of a function.
Ex 1: Please see the graph on the board.
Relative maximum
Rel

5.3 Higher Derivatives, Concavity, and the Second Derivative Test
Higher Derivatives
Studying the derivative has helped us better understand functions. We now know how to find where a function
is increasing/decreasing and where it may have a maximum or a

6.2 Applications of Extrema (new HW list: 5 25 odd; Please read examples 3 & 4 of this section in the textbook)
Procedure: To Solve Applied Extrema Problems
1)
2)
3)
4)
Develop a function that describes the quantity to be maximized or minimized in terms o

6.3 Further Business Applications
Economic Lot Size; Economic Order Quantity; Elasticity of Demand
Economic Lot Size
Our goal here is to determine the number of batches (the production lot size) per year that should be
manufactured in order to minimize to