3.5 Implicit Differentiation
Sometimes we get equations with x and y variables instead of functions. Examples,
x 4 x y y 2 3x y
y
tanx y
1 x2
But we still need to find
dy
. How do we do it?
dx
This is where we use what is known as Implicit Differentiati
4.2 Mean Value Theorem
Rolles Theorem (Chapter 4.2)
Let f be a function that satisfies the following three hypotheses:
1.
f is continuous on the closed interval a, b .
2.
f is differentiable on the open interval a, b .
3.
f a f b
Then, there is a number c
Angles
The radian measure of a positive angle is related to the length s
of an arc, as in the picture below:
y
= s/r radians
Q
r
s
0
P
x
circle of radius r centered at the origin
Converting degrees to radians
Since 360 = 2 radians,
1 =
2
radians =
radian
Functions
Saying that a particular quantity y is a function of another
quantity x means that the value of y is in someway completely
determined by the value of x.
The variable y is called the dependent variable since its value
depends on the value for x.
Slope
The slope of a non-vertical line that passes through two points
(x1 , y1 ) and (x2 , y2 ) is defined to be
m=
y
y2 y1
=
x
x2 x1
y
(x2 , y2 )
(x1 , y1 )
y =y2 y1
x=x2 x1
x
Vertical lines dont have a defined slope.
Horizontal lines have slope 0.
y
(x1
Transformations
Dr. Leslie Walter
Horizontal shifts
If c is a positive real number then the graph of
y = f (x + c)
is obtained by shifting the graph of y = f (x) c units to the
left.
y
y = f (x)
y = f (x + c)
x
moved c units to the left
Example
y
y =1
x
/
Exponential functions
If a 6= 0 is a real number and n is a positive integer then
an = a a a a
| cfw_z
n times
an =
1
aaa a
| cfw_z
n times
If m and n are integers then
am
,
(am )n = amn
an
For the exponent arithmetic to work out, we need to define
am+n
Inverse functions
A function g is called the inverse of a function f if
(g f ) (x) = g (f (x) = x
for all x in the domain of f .
Examples:
1. g (x) = x 2 is the inverse of f (x) = x + 2,
2. g (x) = 3 x is the inverse of f (x) = x 3 ,
3. g (x) = 1/x is the
Combining functions algebraically
If f and g are both functions of x and k is a real number then we
can combine them to get new functions:
(kf ) (x) = kf (x)
(scalar multiple)
(f + g ) (x) = f (x) + g (x)
(sum)
(f g ) (x) = f (x) g (x)
(difference)
(fg )
Sign charts
Determine when the graph of
y = x 2 2x 3
lies below the x-axis.
We need to solve the inequality x 2 2x 3 < 0.
Factor x 2 2x 3 to determine when x 2 2x 3 = 0:
x 2 2x 3 = (x + 1)(x 3)
so x 2 2x 3 = 0 when x = 1 or x = 3.
Because polynomial funct
Math 110.3: Calculus I
2016-2017 Term 1
Section 07
MWF 9:30-10:20am, Phys 107
Dr. Leslie Walter
211 McLean Hall
306-966-6084
[email protected]
Introduction to Calculus
Calculus is the study of continuous functions: if you can draw the
graph of y = f (x
4.9 Antiderivatives
Definition (Chapter 4.9)
A function F is called an antiderivative of f on an interval I if F x f x for all x in I.
In laymans language, an antiderivative is the opposite of derivative.
In other words, we are trying to find the original
3.10
Linear Approximations and Differentials
In both linear approximations (or linearisation) and differentials, the idea is to approximate a function
with its tangent line. Let us assume we wish to find the function value at = 1 . What we need first of
a
3.1 Derivatives of Polynomials and Exponential Functions
Notation:
1.
f x
d
f x
dx
Derivative of a constant function:
d
c 0 ,
dx
2.
c is a constant.
Power Rule:
d n
x nx n 1 ,
dx
3.
n is any real number.
Constant Multiple Rule:
d
cf x c d f x ,
dx
dx
4
3.9 Related Rates
Given a function
y f x
When there is a change in x, let us denote as x , there will be a corresponding change in
y, let us denote as y .
The relationship between x and y is given by
y dy
x dx
dy
y
x
dx
If the rate of change in x is
dx
1.5 Exponential Functions
An exponential function takes the form
f x a x , where the base a 0 .
Notice that this time, the variable x is the exponent.
Question:
Why do we take +ve values for a? Why cant we take non-positive values for a?
Answer:
Although
4.1 Maximum and Minimum Values
More often than not, we want to find the optimum ways of doing things. Some examples:
(Chapter 4.1)
1.
2.
3.
4.
Shape of can that minimises manufacturing costs.
Maximum acceleration of a space shuttle.
Radius of windpipe tha
1.3
New Functions from Old Functions
Translations
This involves the shifting of a graph up or down or left or right.
Vertical and Horizontal Shifts: Suppose we know what the graph of y f x looks
like and suppose we are given c 0 . To obtain the graph of
i
3.3 Derivatives of Trigonometric Functions
Please memorise these trigonometric limits and derivatives where the angle is always
measured in radians.
1.
lim
sin x
1
x
2.
lim
cos x 1
0
x
x 0
x 0
Derivatives of the Trigonometric Functions
3.
d
sin x cos x
dx
2.8 The Derivative as a Function
In Chapter 2.7, one of the formulas we use to calculate the slope of the tangent line to the
curve y f x at x a is
f a lim
h 0
f a h f a
h
Let us look at the following steps:
1.
2.
We generalise point Pa, f a to be Px, f
4.3 How derivatives affect the shape of a graph
When it comes to curve sketching, we can know the orientation of the curve of f by the
values of the first derivative, f , and the second derivative, f .
The Increasing/Decreasing Test (Chapter 4.3)
(a)
(b)
2.1 The tangent and velocity problems
Some definitions: Given a curve, we have the
Secant: The secant line is the line that passes through at least two points on the curve.
y
curve
secant
x
Tangent: The tangent line to a curve is the line that touches the
calculus 1 midterm 1 v6 solutions
Calculus I Midterm 1
October, 2011
2
1. (11 marks total) Answer each of the following.
b) (2 marks) What is the domain of
4
.
3
g ( x) 2 x ?
c) (4 marks) Use Squeeze Theorem to find
12
lim x 6 sin 2 4 .
x 0
x
d) (3 ma