In a related rates problem the idea is to compute the rate of change of one quantity in
terms of the rate of change of another quantity (which may be more easily measured). The
procedure is to find an equation that relates the two qu
FOUR WAYS TO REPRESENT FUNCTION
Functions arise whenever one quantity depends on another.
A function f is a rule that assigns to each element x in a set D exactly one element, called
f x , in e set E .
We usually consider functions for wh
Rates of Change in the Natural and S0cial Sciences
1. The position function of a particle is given by
s t 3 4.5t 2 7t ,
t 0 .
a) When does the particle reach a velocity of 5 m / s ?
b) When is the acceleration 0 ? What is the significance
Derivatives of Logarithmic Functions
log a x 1 ,
x ln a
ln u 1 du
ln x 1
ln g x g x
Differentiate the function.
f x log 5 xe x
b) h x ln x x 2 1
Sec. 3.10 Linear Approximations and Differentials
f x f a f a x a
Is called linear approximation or tangent line approximation of f at a .
The linear function whose graph is this tangent line, that is
L x f a f a x a
The greatest challenge in optimization problem is setting up the function (the goal
function) that is to be maximized or minimized.
STEPS IN SOLVING OPTIMIZATION PROBLEMS
1. Understand the Problem The first step is t
LAWS OF EXPONENTS If a and b are positive numbers and x and y are any real numbers, the
1. a x y a x a y
2. a x y
a x b y
Sketch the graph of the function y 4 x 3 and y 4 x 3 . Iden
Sec. 4.7 (part 2)
Find the point on the line y 4 x 7 that is closest to 2,5 . (Answer:
A right circular cylinder is inscribed in a cone with height h and base radius r . Find the
largest possible volume of such a cylinder. (Answe
Exponential Growth and Decay
In many natural phenomena, quantities grow or decay at a rate proportional to their size. For
y f t is the number of individuals in a population of animals or bacteria at time t , then it
MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS
A Catalog of Essential Functions
1. A linear function f x mx b
2. A polynomial of degree n : P x a n x n a n 1 x n 1 a 2 x 2 a1 x a 0
3. Power functions: f x x a , where a is a constant.
.t .MAT.H=1A . . . . . . TEEFE ,
- solutions. t -:
Directions: Show all work to receiVe-a partial credit for ircorr'ect solutith7éEEHE'tGWEd'it'foFCWWWM"'
tél 'Acvlinqrical can Without-a top is mad-etdCbntain-100OCm?-éf'liqt1id. Find the
Math 1A-27 (01223) E31: Calculus I, Winter Quarter 2015
Mon-Thur: 1:45 to 2:35
Prerequisite: Passing grade (C or better) in Math 49B and qualifying score on Cal
The Chain Rule
THE CHAIN RULE If g is differentiable at x and f is differentiable at g x , then the composite
function F f g is differentiable at x and F is given by product
F x f g x g x .
In Leibniz notation, if y f u and u g x are both
Find y by implicit differentiation.
Solve the equation explicitly for y and differentiate to get y in terms of x
Check that your solutions to parts (a) and (b) are consistent by substituting the
In this section we study a numerical method, called Newtons method or the Newton-Raphson
method, which is a technique to approximate the solution to an equation f x 0 . Essentially it
uses tangent lines in place of the grap
Geometry of the First and Second Derivatives
Attach this sheet as a cover sheet.
Specific Objectives The student will get information about the graph of f from the graphs of
the first derivative function f . With this information and an ini
Inverse Functions and Logarithms
Definition 1 A function f is called a one-to-one if it never takes on the same value twice:
f x1 f x 2
x1 x 2
HORIZONTAL LINE TEST A function is one-to-one if and only if no horizontal l
A physicist who knows the velocity of a particle might wish to know its position at a given time.
An engineer who can measure the variable rate at which water is leaking from a tank wants to
know the amount leaked over cer
New Functions from Old Functions
Sketch the graph of y x .Use transformations to graph y x 2 , y x 2 , y x
, y 2 x .
Sketch the graph of the function y x 4 .
COMBINATIONS OF FUNCTIONS
g x f x g x
fg x f x g x
g x f x g
The Limit of a Function
lim x a f ( x) L
the limit of f x , as x approaches a , equals L
if we can make the value of f x arbitrary close to L (as close to L as we like) by taking x sufficiently
close to a (o
CALCULATING LIMITS USING LAWS OF LIMITS
Suppose that is a constant and the limits
Ex 1 Use the Limit Laws and the graphs of and in Figure 1 to evaluate the following limits if
The Precise Definition Of a Limit
To motivate the precise definition of a limit, lets consider the function
2 x 1 if x 3
if x 3
Intuitively, it is clear that when x is close to 3 but x 3 , then f x is close to 5 , and so
lim x 3 f
We noticed in Sec. 2.3 that the limit of a function as x approaches a can often be found
simply by calculating the value of the function at a . Functions with this property are called
continuous at a . We will see that the math
Sec. 2.6 LINITS AT INFINITY: HORIZONTAL ASYMPTOTE
Let f be a function defined on some interval a, . Then
lim x f x L
means that the values of f x can be made arbitrary close to L by taking x sufficiently large.
Similarly, we define
DERIVATIVES OF POLYNOMIAL AND EXPONENTIAL FUNCTIONS
1. Derivative of a constant function :
2. The Power Rule:
c 0 .
x nx n 1
3. The Power Rule(general version):
x nx n 1 if n any real number.
Math 1A Handout for Final Exam
The Final Exam is two parts. You are not allowed to use your calculator for the
first part of the exam. When you are done with part 1, you turn it in and receive
part 2; you may then use your calculator. You may use as much
We noticed in Sec. 2.3 that the limit of a function as x approaches a can often be found simply
by calculating the value of the function at a . Functions with this property are called continuous at a .
We will see that the math
Hard Determinism has a number of profound consequences. It puts into
doubt our hopes for the future and how we consider the morality of others.
Determinism means that were mistaken to praise some people for doing
good and blame others for
The dictionary definition of empathy is "the ability to
understand and share the feelings of another". One of
the examples I can think of Noddings' idea that human
beings perform ethical actions because they possess a
"caring response" is that of M