Math 124 Final Information Guide



Math 124 Final Information Guide

Chapter 1; Library of Functions

Functions and Change

Exponential Functions

New Functions From Old

Logarithmic Functions

Trigonometric Functions

Powers, Polynomials, and Rational Functions7

Chapter 2; The Derivative

2.1 How to we measure speed?

2.2 The Derivative at a Point

2.3 The Derivative Function

2.4 Interpretation of the Derivative

2.5 The Second Derivative

2.6 Differentiability

Chapter 3; Short Cuts to Differentiation

3.1 Powers and Polynomials

3.2 The Exponential Function

3.3 The Product and Quotient Rule

3.4 The Chain Rule

3.5 The Trigonometric Functions

3.6 The Chain Rule and Inverse Functions

3.7 Implicit Functions

3.8 Hyperbolic Functions

3.9 Linear Approximations and the Derivative

Chapter 4; Using the Derivative

4.1 Using the First and Second Derivatives

4.2 Family of Curves

4.3 Optimization

4.4 Applications to Marginality

4.5 Optimization and Modeling

4.6 Rates and Related Rates

4.7 L’Hopital’s Rule, Growth, and Dominance

4.8 Parametric Functions

Chapter 5; The Definite Integral

5.1 How do we Measure Distance Traveled?

5.2 The Definite Integral

5.3 The Fundamental Theorem and Interpretations

5.4 Theorems about Definite Integrals

Chapter 6; Constructing Antiderivatives

6.1 Antiderivatives Graphically and Numerically

6.2 Constructing Antiderivatives Analytically

6.3 Differential Equations

6.4 Second Fundamental Theorem of Calculus

6.5 The Equations of Motion

Chapter 7

7.1 Integration by Substitution

Equations

Chapter 1

Linear Function

y = f(x) = b + mx

m = [pic]

Directly Proportional

y = kx

Exponential Growth; base “a” and “e”

P = P0at Q = Q0et

Exponential Decay: base “a” and “e”

P = P0a-t Q = Q0e-t

Graphing

• Multiplying a function by a constant, c, stretches the graph vertically (if c > 1) or shrinks the graph vertically (if 0 0 and to the left if c < 0 |

|f(kx) |The graph of f scaled horizontally be a factor of k |

|f(x)+b |The graph of f moved vertically b units |

The period of

y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d

where k is a non-zero real number, is 2[pic]/|k|

The amplitude of

y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d

where k is a non-zero real number, is |A|

Power Functions

f(x) = kxp

Horizontal Asymptote

Let y = f(x) be a function. Suppose that L is a number such that whenever x is large, f(x) is close to L and suppose that f(x) can be made as close as we want to L by making x larger.

Then we say that the limit of f(x) as x approaches infinity is L and we write

[pic]

Vertical Asymptote

Let f be a function which is defined on some open interval containing “a” except possibly at x = a. We write

[pic]

if f(x) grows arbitrarily large by choosing x sufficiently close to “a”.

Intermediate Value Theorem: Let f be a function which is continuous on the closed interval [a, b]. Suppose that d is a real number between f(a) and f(b); then there exists c in [a, b] such that f(c) = d.

Definition of a limit

|The limit of f(x) as x approaches a is L |

|[pic] |

|if and only if, given ε > 0, there exists δ > 0 such that 0  ................
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