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 y m = x Directly Proportional y = kx Exponential Growth; base “a” and “e” t t P = P0a Q = Q0e

Exponential Decay: base “a” and “e”

-t -t P = P0a Q = Q0e

Graphing  Multiplying a function by a constant, c, stretches the graph vertically (if c > 1) or shrinks the graph vertically (if 0

1. loga (uv) = loga u + loga v 1. ln (uv) = ln u + ln v

2. loga (u / v) = loga u – loga v 2. ln (u / v) = ln u – ln v n n 3. loga u = n loga u 3. ln u = n ln u Change of Base Formula

m > 0 and b > 0

Arc Length = s = rradiusӨ Trigonometric Functions and Modifiers Modified function How the graph is obtained Af(x) The graph of f with y values scaled by a factor of A The graph of f shifted |c| units to the right if c > 0 and to the left if c < f(x-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 /|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

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

if and only if, given  > 0, there exists  > 0 such that 0 < |x - a| <  implies that |f(x) - L| < .

Properties of limits lim b = b xc lim x = c xc lim x n = c n xc lim b[f(x)] = b lim f(x) xc xc lim [f(x) + g(x)] = lim f(x) + lim g(x) xc xc xc lim [f(x)g(x)] = [ lim f(x)][ lim g(x)] xc xc xc lim f(x) = lim f(x) xc g(x) x  c assuming lim g(x)  0 lim g(x) xc xc Definition of Continuity

Chapter 2 Change _ in _ Position Average Velocity = Change _ in _Time Instantaneous Velocity

Derivative/ Instantaneous Rate of Change

If f’(x) is positive the function is increasing over the interval. If f’(x) is negative the function is decreasing over the interval If f’’(x) is positive on an interval then f’(x) is increasing, so the graph of f(x) is concave up If f’’(x) is negative on an interval then f’(x) is decreasing, so the graph of f(x) is concave down

Functions that are not differentiable have/are: Cusps (sharp corners), piecewise, or oscillate to infinity at a point

Chapter 3

Derivative Short-cuts (10)

(21) (22)

Rules of differentiation

Sum Rule: (f+g)'=f'+g' Constant Multiple Rule: (cf)'=cf' Product Rule: (fg)'=f'g+fg'

Quotient Rule:

Chain Rule: f(g(x))' = f'(g(x))g'(x)

or , where y=f(u) and u=g(x)

Derivative of an inverse function

Chapter 4 L’Hopital’s Rule

Parametric Equations for a Straight line

x = x0 + at and y = y0 + at m = b/a Parametric- Instantaneous speed  dx   dy  v = [  2 +   2] (1/2)  dt   dt  Chapter 5

Fundamental Theorem of Calculus Theorem. Let f be a function which is continuous on the interval [a, b].

Second fundamental theorem of Calculus

(2) Then

Integration table of basic functions 1.

3. ; in particular,

4. 5. 7.

Properties of integration 1.

3. ; in particular, 4. 5. 7.

Integration by parts