Calculus and Vectors How to get an A+
5.4 5.5 Derivative of Trigonometric Functions A Review of Trigonometric Functions
sin x : R [1,1] cos x : R [1,1] sin( x + / 2) = cos x sin( x + 2 ) = sin x cos(
Calculus and Vectors How to get an A+
5.4 5.5 Derivative of Trigonometric Functions A Review of Trigonometric Functions
sin x : R [1,1] cos x : R [1,1] sin( x + / 2) = cos x sin( x + 2 ) = sin x cos(
Calculus and Vectors How to get an A+
5A Derivative of Logarithmic Function A Review of Logarithmic Function y = b x x = log b y y = f ( x) = log b x, b > 0, b 1, x > 0
log b ( xy ) = log b x + log b
Calculus and Vectors How to get an A+
5A Derivative of Logarithmic Function A Review of Logarithmic Function y = b x x = log b y y = f ( x) = log b x, b > 0, b 1, x > 0
log b ( xy ) = log b x + log b
2.4 Quotient Rule A. Quotient Rule (Product Rule) If f and g are differentiable f at x and g ( x) 0 , then so is and (in Lagrange g notation) f f ' ( x) g ( x) f ( x) g ' ( x) ' ( x ) = , g ( x) 0 g [
Calculus and Vectors How to get an A+
2.4 Quotient Rule A Quotient Rule If f and g are differentiable at x and g ( x) 0 then so f is and: g
f ( x) f ' ( x ) g ( x ) f ( x) g ' ( x ) g ( x) = [ f ( x)
Calculus and Vectors How to get an A+
3.2 Maximum and Minimum on an Interval. Extreme Values A Global Maximum A function f has a global (absolute) maximum at x = c if f ( x) f (c) for all x D f .
f (c
1.5 Tangent and Normal Lines A. Lines (Slope y-intercept Equation) The equation of a non vertical line in slope y-intercept form is: y = mx + b where m is the slope of the line and b is the yintercept
Calculus and Vectors How to get an A+
3.1 Higher Order Derivatives. Velocity and Acceleration A Higher Order Derivatives Let consider the function y = f ( x) . The first derivative of f or f prime is:
1.2 Limits (I) A. Overview (Limit) The concepts of limit is related to the behaviour of a function f ( x) in the neighbourhood of a number x = a . Consider a function f ( x) defined on a neighbourhood
1.6 Rates of Change A. Rate of Change Consider a function y = f ( x) and two points (Finite Differences) The following table allows the calculation of the rate of change for all consecutive ordered pa
3. Optimization Algorithm for Solving Optimization Problems 1. Understand the problem 2. Draw a diagram 3. Assign variables to quantities involved 4. Write relations between these variables 5. Identif
3.2 Minimum and Maximum Values A. Minimum and Maximum Values (Local Maximum) A function f has a local (relative) maximum at x = c if f ( x) f (c) when x is sufficiently close to c (on both sides of c
Calculus and Vectors How to get an A+
3.3 3.4 Optimization A Algorithm for Solving Optimization Problems 1. Read and understand the problems text. 2. Draw a diagram (if necessary). 3. Assign variables
3.4 Asymptotes A. Vertical Asymptote (Infinite Limits) If the values of f ( x) can be made arbitrarily large by taking x sufficiently close to a with x < a , then lim f ( x) = .
xa
Note. A Horizontal
Calculus and Vectors How to get an A+
1.6 Continuity A Continuity A function y = f ( x) is continuous at a number a if lim f ( x) = f (a )
x a
Note: A function is continuous if the graph can be drawn
Calculus and Vectors How to get an A+
1.2 The Slope of the Tangent A Lines Ex 1. The equation of the line L1 is: 2 x 3 y + 6 = 0 . a) Find the slope of the line L1 .
b) Find the equation of the line L
Calculus and Vectors How to get an A+
2.1 Derivative Function A Derivative Function Given a function y = f ( x) , the derivative function of f is a new function called f ' (f prime), defined at x by:
Calculus and Vectors How to get an A+
2.2 Derivative of Polynomial Functions A Power Rule Consider the power function: y = x n , x, n R . Then:
( x )' = nx dn x = nx n 1 dx
n n 1
Ex 1. For each case,
Calculus and Vectors How to get an A+
1.4 Limit of a Function A Left-Hand Limit If the values of y = f ( x) can be made arbitrarily close to L by taking x sufficiently close to a with x < a , then: li
1.4 Continuity A. Overview (Continuity) The concept of limit continuity is related to the behaviour of a function f ( x) in the neighbourhood of a number x = a including this number. Example. Consider
Calculus and Vectors How to get an A+
1.5 Properties of Limits A Limits Properties We assume that lim f ( x) and lim g ( x) exist. Then:
x a x a
Ex 1. Given lim f ( x) = 2 and lim g ( x) = 1 , use the
Calculus and Vectors How to get an A+
1.3 Rate of Change A Average Rate of Change y = f ( x), y1 = f ( x1 ), y 2 = f ( x2 ) x = x 2 x1 ( change in variable x ) y = y 2 y1 ( change in variable y ) The
Calculus and Vectors How to get an A+
1.1 Radical Expressions: Rationalizing Denominators - Handout
Ex 1. Simplify:
A Radicals
a a =a
n
a)
( a) = a
n
m
( a )=
m
an
3 3
b) ( 5 ) 3
n
= (n a ) m
Note: If
Calculus and Vectors How to get an A+
3.3 3.4 Optimization A Algorithm for Solving Optimization Problems 1. Read and understand the problems text. 2. Draw a diagram (if necessary). 3. Assign variables
Calculus and Vectors How to get an A+
3.2 Maximum and Minimum on an Interval. Extreme Values A Global Maximum A function f has a global (absolute) maximum at x = c if f ( x) f (c) for all x D f .
f (c
2.5 Chain Rule A. Chain Rule (Chain Rule) If g is differentiable at x and f is differentiable at g ( x) , then the composition
dy dy du to find the = dx du dx derivative of y = f ( g ( x) .
2. For eac
2.2 Basic Differentiation Rules A. Pow er Rule C. More Basic Differentiation Rules (Power Function) The power function is defined by: f ( x) = x , The domain and the range of the power function depend
Calculus and Vectors How to get an A+
3.1 Higher Order Derivatives. Velocity and Acceleration A Higher Order Derivatives Let consider the function y = f ( x) . The first derivative of f or f prime is:
Calculus and Vectors How to get an A+
2.1 Derivative Function A Derivative Function Given a function y = f ( x) , the derivative function of f is a new function called f ' (f prime), defined at x by: