# MATH 2A Study Guide - Fall 2018, Comprehensive Midterm Notes - Graph Of A Function, Integer, TranscendentalsPremium

by OC2512313

Department

MathematicsCourse Code

MATH 2AProfessor

Liu, WStudy Guide

MidtermThis

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MATH 2A

Single-Variable Calculus

Midterm

Fall 2018

Prof. Liu, W

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MATH 2A LECTURE 2- EXPONENTIAL FUNCTIONS

Exponential Functions

● f(x) = ax

● A is called the base and a > 0

● x=n. N is a positive integer.

● An = a * a * a

Rules of Exponents

1) Ax+y = Ax * Ay : Adding exponents is the same as multiplying the exponential functions

2) Ax-y = Ax / Ay : Subtracting exponents is the same as dividing the exponential functions

3) Axy = (Ax)y

Graphing

- a>1

- Increasing

- Domain: all real numbers (-∞ , ∞ )

- Ax is not equal to 0 and is not negative

https://prep.math.lsa.umich.edu/cgi-bin/pmc/crtopic?sxn=14&top=1&crssxn=prep

- a>1

- (( )x = bx)

b

1

- The base graph is being reflected on the y-axis.

- Decreasing

- Domain: all real numbers (-∞ , ∞ )

- Range: (0,∞)

https://prep.math.lsa.umich.edu/cgi-bin/pmc/crtopic?sxn=14&top=1&crssxn=prep

Example: Sketch the graph of y = 5 -3 * 2x

1) Start with basic graph of 2x

2) Shift the graph UP THREE spaces to account for

the times 3.

3) REFLECT the graph to account for the negative

sign in front of the 3.

4) Shift the graph UP FIVE spaces to account for the

5.

5) The graph should look like this in the end.

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Example: Find the exponential function whose graph passes through (1,6) and (3,24)

f(1) = 6 f(3) =24

y = f(x) = cax

c*a = 6 c= a

6

c* a3 = 24 * a3 = 24

6

a

6a2 = 24

A2 = 4

a=2, c = 3 so f(x) = 3 * 2x

The Number E

● Slopes of tangent lines at (0,1) to exponential graph will cover all possible values.

● There is an exponential where the slope is 1. This will have base e (ex).

1.5 INVERSE FUNCTIONS

● Function from A → B...create a function g

● Essentially, you are switching x and y in order to make an inverse

Ex:

f(1) = 3 and g(3)= 1

f(x) = y → g(y)= x

Problem: f(1,2,3) → (1,2)

f(1) = 1 g(1,2) → (1,2,3)

f(2) = 2 g(1) = 1

f(3) = 2 f-1 (2) → (2,3)

If f has an inverse, f must only take each value once. This is called a one to one function.

Horizontal Line Test

● For a one to one function, every horizontal line must touch the

graph of f at least once.

● https://www.chilimathwords.com/math-words-starting-with-letter-h/horizontal-line-test/

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