1. Calculus I Hughes-Hallett
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Calculus I Hughes-Hallett
Math 131
Br. Joel Baumeyer
Christian Brothers University

2. Function: (Data Point of View)
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Function: (Data Point of View)
One quantity H, is a function of another, t, if each value of t has a unique value of H associated with it. In symbols: H = f(t).
We say H is the value of the function or the dependent variable or output; and
t is the argument or independent variable or input.

3. Working Definition of Function: H = f(t)
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Working Definition of Function: H = f(t)
A function is a rule (equation) which assigns to each element of the domain (independent variable) one and only one element of the range (dependent variable).

4. Working definition of function continued:
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Working definition of function continued:
Domain is the set of all possible values of the independent variable (t).
Range is the corresponding set of values of the dependent variable (H).

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Questions? 11

6. General Types of Functions (Examples):
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General Types of Functions (Examples):
Linear: y = m(x) + b; proportion: y = kx
Polynomial: Quadratic: y =x2 ; Cubic: y= x3 ; etc
Power Functions: y = kxp
Trigonometric: y = sin x, y = Arctan x
Exponential: y = aebx ; Logarithmic: y = ln x

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Graph of a Function:
The graph of a function is all the points in the Cartesian plane whose coordinates make the rule (equation) of the function a true statement.

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Slope
m - slope :
b: y-intercept
a: x-intercept .

9. 5 Forms of the Linear Equation
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5 Forms of the Linear Equation
Slope-intercept: y = f(x) = b + mx
Slope-point:
Two point:
Two intercept:
General Form: Ax + By = C

10. Exponential Functions: If a > 1, growth; a<1, decay
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Exponential Functions: If a > 1, growth; a<1, decay
If r is the growth rate then a = 1 + r, and
If r is the decay rate then a = 1 - r, and

11. Definitions and Rules of Exponentiation:
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Definitions and Rules of Exponentiation:
D1:
D2:
R1:
R2:
R3:

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Inverse Functions:
Two functions z = f(x) and z = g(x) are inverse functions if the following four statements are true:
Domain of f equals the range of g.
Range of f equals the domain of g.
f(g(x)) = x for all x in the domain of g.
g(f(y)) = y for all y in the domain of f.

13. A logarithm is an exponent.
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General Rules of Logarithms: log(a•b) = log(a) + log(b) log(a/b) = log(a) - log(b)

15. e = 2.718281828459045... Any exponential function
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e =
Any exponential function
can be written in terms of e by using the fact that
So that

16. Making New Functions from Old
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Making New Functions from Old
Given y = f(x):
(y - b) =k f(x - a) stretches f(x) if |k| > 1
shrinks f(x) if |k| < 1
reverses y values if k is negative
a moves graph right or left, a + or a -
b moves graph up or down, b + or b -
If f(-x) = f(x) then f is an “even” function.
If f(-x) = -f(x) then f is an “odd” function.

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Polynomials:
A polynomial of the nth degree has n roots if complex numbers a allowed.
Zeros of the function are roots of the equation.
The graph can have at most n - 1 bends.
The leading coefficient determines the position of the graph for |x| very large.

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Rational Function: y = f(x) = p(x)/q(x) where p(x) and q(x) are polynomials.
Any value of x that makes q(x) = 0 is called a vertical asymptote of f(x).
If f(x) approaches a finite value a as x gets larger and larger in absolute value without stopping, then a is horizontal asymptote of f(x) and we write:
An asymptote is a “line” that a curve approaches but never reaches.

19. Asymptote Tests y = h(x) =f(x)/g(x)
Vertical Asymptotes: Solve: g(x) = 0
If y    as x  K, where g(K) = 0,
then x = K is a vertical asymptote.
Horizontal Asymptotes:
If f(x)  L as x   then y = L is a vertical asymptote. Write h(x) as:
, where n is the highest power of x in f(x) or g(x).

20. Basic Trig radian measure: q = s/r and thus s = r q,
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Basic Trig
radian measure: q = s/r and thus s = r q,
Know triangle and circle definitions of the trig functions.
y = A sin B(x - j) + k
A amplitude;
B - period factor; period, p = 2p/B
j - phase shift
k (raise or lower graph factor)

21. Continuity of y = f(x)
A function is said to be continuous if there are no “breaks” in its graph.
A function is continuous at a point x = a if the value of f(x)  L, a number, as x  a for values of x either greater or less than a.

22. Intermediate Value Theorem
Suppose f is continuous on a closed interval [a,b]. If k is any number between f(a) and f(b) then there is at least one number c in [a,b] such that f(x) = k.