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( x + 2 ) = cos x sin( x + ) = sin x
Ex 1. Compute the fo
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( x + 2 ) = cos x sin( x + ) = sin x
Ex 1. Compute the fo
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 x x log b = log b x log b x y
log b x = n log b x log a
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 x x log b = log b x log b x y
log b x = n 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 [ g ( x)] 2
f f ' g fg ' ' = g g2 or (in Leibniz notat
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)]2 f f ' g fg ' = g g2 d d g ( x) f ( x ) f ( x) g ( 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) is called the global (absolute) maximum value. (c, f
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. (Slope Formula) If two points P1 ( x1 , y1 ) and
P(a,
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: dy f ' ( x) = y ' = dx The second derivative of f or f
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 of the number x = a , eventually not defined at x = a
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 pairs (process called numerical derivative):
x x y y
P1 (
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. Identify the variable that is minimized or maximized (the depe
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 ). f (c) is called local (relative) maximum value and (
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 to the quantities involved and state restrictions acco
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 asymptote may be crosses or touched by the graph of the
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 without lifting the pen from paper.
Note: A function is
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 L2 , that is parallel to the line L1 and passes through
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: f ( x + h) f ( x ) f ' ( x) = lim h0 h B Differentiabil
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, differentiate. a) (1)' b) ( x)' c) ( x 5 )'
1 d) x
'
So
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: lim f ( x) = L
x a
Ex 1. Use the function y = f ( x) defi
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 the function f ( x) defined by the following graph: Ex
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 limits
x 3 x 3
1. lim k = k
x a
properties to find lim
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 Average Rate of Change (ARC) in y variable over the int
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 n is even, then a 0 for
n
c) (3 7 ) 5
a.
B Rationalizi
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 to the quantities involved and state restrictions acco
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) is called the global (absolute) maximum value. (c, f
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 each case, use
( f o g )( x) = f ( g ( x) is differentiabl
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 on the values or the exponent . (Power Rule) The deriv
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: dy f ' ( x) = y ' = dx The second derivative of f or f
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: f ( x + h) f ( x ) f ' ( x) = lim h0 h B Differentiabil