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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