CALCULUS
By: Vanessa F.

EXPONENTIAL & TRIGONOMETRIC
WITH DERIVATIVES

Exponential

f(x) = e^x, then f'(x) = e^x

g(x) = e^h(x), then g'(x) = e^h(x) . h'(x)

Logarithmic

The function for this is log power over base

e^in x= x

Derivatives

If f(x) = b^x then f'(x). b^x . In b

If g(x) = b^h(x) then g'(x) = b^h(x) . In b . h'(x)

Trigonometric

If f(x)= sinx then f'(x) = cos x

If f(x) = cos x then f'(x) = -sinx

If f(x) = tan x then f'(x) = sec^2 x

also, tanx = sinx/cosx

ex, y= (sinx + tan x)^4
dy/dx= 4 (sinx + tanx)^3 . (cos + sec^2x)
= 4 ( cosx + sec^2 x ) ( sinx + tanx)^3

INTRODUCTION

Radical Expressions

√a x √a = √a2 = a

(a - b) (a+b) = a2 - b2

( √m - √n ) ( √m + √n ) = √m2 - √n2 = m - n

If the expression is the
numerator, rationalise by
multiplying the numerator
and denominator by its
conjugate.

If the expression is in the
denominator, rationalise by
multiplying the numerator and
denominator by its conjugate.

when there is no denominator,
simplify.

MINOMIAL

Slope of a Tangent

lim f ( a + h ) - f (a) / h

h→0

Rates of Change

Instantaneous Velocity

Instantaneous Rate of Change

< Slope of the tangent

Average Rate of Change

< Slope of the secant

Limits

Indeterminate form 0/0

ONE SIDED LIMITS if it is a absolute value

CHANGE OF VARIABLE if there a cube roots or other

RATIONALISING if there is a square root

FACTORING if there is a restriction

Limit exists if both one-sides limits
are the same

Limit D.N.E if the one-sides limits
are not the same

The number L = the limit of the function
y = f(x) as x approaches the value 'a'

Continuity

Function is continuous when

limit of f(x) = f(a)

both side limits are equal

f(a) is defined

SECOND DERIVATIVES

Optimization Problems

Solving Steps:
- Define variables
- Make an equation that defines the question
- Apply any substitutions if needed
- Set the derivative to 0
- Find the critical values
- Find the test values
- Find the domain and range

Ex, A piece of cardboard, 80 cm by 40 cm,
is used to make an open-top rectangular box.
Find the dimensions for the box with the largest volume.

First I would draw a diagram
to better understand the question.
Then I would let x represent the width
of the removed square.
For this particular question, I would use
the formula V = (L)(W)(h)
I then plug the number given in the question
into this formula and find the first derivative.
I ten set the derivative to 0. Once i find the
critical value, I find my test values and the dimensions.

Extreme Values

Critical Values

f'(c) = 0 or f'(c) D.N.E

Minimum

Local min
- the second lowest point on a
graph

Absolute min
- the highest point on the graph

Maximum

Local max
- the second highest point on a
graph

Absolute max
- the highest point on a graph

To find the max and min on an interval:
- Differentiate
- Set f'(x) = 0
- Solve of x in f'(x) = 0 or where f'(x) D.N.E
- Test values
- Compare the tested values

Velocity and Acceleration

The function notation for the
second derivative is considered
as, f''(x)

To find the second derivative,
You have to find the first derivative
of the function being given.

ex, f(x) = 2x^2 + 5x + 6
f(x) = 4x + 5
f''(x) = 4
Therefore, the second derivative is 4.

Velocity

v(t) = s'(t)

The instantaneous rate of change
of the position function s(t)

-If the object is moving up or right,
the v(t) is > 0
-If the object is moving down or left,
the v(t) is < 0
- If the object is at rest, v(t) = 0

Acceleration

a(t) = v'(t) = s''(t)

The instantaneous rate of change
of the velocity with respect to time

This is known as the second derivative
of the s(t) and first derivative of v(t)

FIRST DERIVATIVES

Composite Functions

Defined as (f. g) = f(g(x))

Chain Rule

h'(x)= f'(g(x)) . g'(x)

Lebniz Notation
dy/dx = dy/du - du/dx

Quotient Rule

f'(x) = p'(x) . q(x) - p(x) . q'(x)/ [q(x)]^2

Product Rule

f'(x) = p'(x) . q(x) + p(x) . q'(x)

Power of a Function Rule

f'(x) = n[g(x)] ^n-1 . g'(x)

Polynomial Functions

Lebniz Notation

f(x) = f'(x)

y' = dy/dx

Sum and Differences Rule

f(x) = p(x) +- q(x), then f'(x) = p'(x) +- q'(x)

Constant Multiple Rule

f(x) = k [g(x)], then f'(x) = K [g'(x)]

Power Rule

f(x) =x^n, then f'(x) = nx^n-1

Constant Function Rule

f(x) = k, then f'(x) = 0

Horizontal Tanget

When the slope of a tangent is at 0

The Function

f'(x)

Function f(x) is differentiable
only if f'(a) exists

Its not differentiable if f(x) is
- Cusp
- Vertical Tangent
- Discontinuity
- Corner

f'(x) is always one degree
less than f(x)

f'(x) = lim f ( a + h ) - f (a) / h

h→0

CURVE SKETCHING

Increasing and Decreasing
Functions

Function is increasing if,
f(x1) < f(x2) when x1 < x2

Function is decreasing if,
f(x1) > f(x2) when x1 < x2

To find the values on inc/dec,
use a interval chart

First Derivative Test

If f'(x) changes from + to - , f has a local max

If f'(x) changes from - to +, f has a local min

If f'(x) sign does not change then the max and min D.N.E at c

Critical Points

A critical value is also
known as c

if the value is f(x) and f'(c) = 0 or f'(c) it is undefined

To find the critical values of a function,
Make the first derivative of the function
equal to 0.

VA, HA and OA

Vertical Asymptotes (VA)

Domain restriction is, x is not = to ...

Is determined by factoring the numerator
and denominator of rational expressions

Horizontal Aysmptotes (HA)

To determine limits as x approaching
infinity of a rational function, divide the
numerator and denominator by the highest
degree then simplify

Oblique Asymptotes (OA)

If the degree on the numerator
is one degree higher than the
denominator

To determine OA,
use long division

Concavity/ P.O.I

If f''(x) > 0 then f(x) is concave up
( also known as a curve facing up)

If f''(x) < 0 then f(x) is concave down
( also known as a curve facing down)

The Point of Inflection is defined if
f''(x) changes sign at x.

Second Derivative Test

If f'(c) + 0 and f''(c) > 0, then local min a c

If f'(c) and f''(c) < 0, then local max at c