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A RainsfordStudy Guide

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01:640:135

Calculus I

Fall 2017

Term Test 1

Prof: RAINSFORD, A

Exam Guide

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Pre-Calculus Review

Composition of Functions

f(x) = x/(x 1) g(x) = 3x+5x 1

Find both composite functions:

Composite= taking the function of another function

g(f(x)) = 3x

x1 + 5 x

x11

fg(x)= 3x+5x 1 / (3x+5x 1 1)

● By taking g(f(x)), you are replacing the x in g(x) with the function f(x)

● By taking f(g(x)), you are replacing the x in f(x) with the function g(x)

Special Domains

● √0=0

● Ln (positive)

○ Cannot be =0

● An even root must be zero or positive (greater than or equal to zero)

○ Therefore, you can set the contents of a square root greater than or equal to

zero and solve (1st step in example 1 below)

● Functions with Real Domains

○ Polynomial functions (many x terms to different powers +- constant)

○ Exponential function

○ sine/ cosine functions (their domain always lies between -1 and 1 think unit

circle)

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Examples

1.) f(x) = (x + 3)(x 3)

(x 3) (x + 1) 0

3 and -1 are the critical points- plot them on the number line

Plug in any value less than -1, between -1 and 3, and greater than 3 to see if domain

exists (+) or not (-)

+ - +

←---------------------------->

-1 3

Domain: (-∞, -1] U [3, ∞)

Brackets used instead of parentheses because those values would make the inequality

equal to 0 and 0 is included in the domain since the square root can be positive or equal

to zero

2.) g(x) = (x 1)/ (x 3)

Same terms as #1 → same + and - intervals on the number line!!

However, x ≠ 3 because that would make the denominator=0, which would make the

value in the square root undefined, therefore in the number line from example 1, the

point x=3 would be represented by an open circle to show that it is not included in the

domain

Parenthesis after 3 because 3 is NOT included in the domain

Domain: (-∞, -1] U (3, ∞)

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