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01:640:135 Midterm: Term Test 1 - Fall 2017Premium


Department
Mathematics
Course Code
01:640:135
Professor
A Rainsford
Study Guide
Midterm

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Rutgers University
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 #1same + 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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