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


Department
Mathematics
Course Code
MATH 2A
Professor
Liu, W
Study Guide
Midterm

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University of California - Irvine
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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