math_113_20140916033120 Flashcards
The Squeeze Theorem
Don’t know the definition, but you’re supposed to find the range of the function (usually sin or cos, so between -1 and 1) so it is in the format -1
Intermediate Value Theorem
If f(x) is continuous on the closed interval [a,b], let f(a)
Extreme Value Theorem
If f is continuous on a closed and bounded interval [a,b], then f surely attains both an absolute max and an absolute min on [a,b].
Rolle’s Theorem
If f(x) is continuous on [a,b], if f(x) is differentiable on (a,b), and if f(a)=f(b), then there exists a value c in (a,b) such that f’(c)=0.
Mean Value Theorem
If f(x) is continuous on [a,b], and if f(x) is differentiable on (a,b), then there exists a value c in (a,b) where f’(c)=(f(b)-f(a))/(b-a).
Fundamental Theorem of Calculus I
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Fundamental Theorem of Calculus II
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sinx
cosx
cosx
-sinx
tanx
sec^2x
cscx
-cscxcotx
secx
secxtanx
cotx
-csc^2x
Linerization
f(x)=L(x)=f(a)+f’(a)(x-a)
Reimann Sums
A=limit as x approaches infinity of the sum (n on top i=1 on bottom) f(xi)delta(x) where delta(x)=(b-a)/n and xi=a+idelta(x)
First Principles
(f(x+h)-f(x))/h
Steps for Curve Sketching
DISA ILCSDomain, Intercepts, Symmetry, Asymptotes, Intervals of Increase and Decrease, Local Max/Min, Concavity and Inflection Points, Sketch.
Sum (n on top k=1 on bottom) i
(n(n+1))/2
Sum (n on top k=1 on bottom) i^2
(n(n+1)(2n+1))/6
Absolute Maximum or Minimum
If the local max/mins are true for all values of x, then f(c) is an absolute max/min.
Local Minimum
f(x) > or equal to f(c) for all x in some interval.
Local Maximum
f(x)
Critical Number
Place in the domain where f’(x) =0 or where f’(x) DNE.
Increasing Function
If the function is rising to the right.
Decreasing Function
If the function is falling to the left.
Concave Up Function
If the graph lies above the tangent line.
Concave Down Function
If the graph lies below the tangent line.
Antiderivative
F(x) is an antiderivative of f(x) if F’(x)=f(x).
Indefinite Integral
The set of all antiderivatives of a function f(x).
Integrand
The function in an integral.
Inflection Point
A point where the graph of f(x) has a tangent line and changes from concave down to up or up to down.
Real Numbers
Numbers that can be expressed as decimals.
Set
A collection of elements.
U
Union
U (upside down)
Intersection
Interval
Set of numbers with no holes.
Abs(x)
Two situations, one is greater than or equal to zero and is x. Other one is less than zero and -x.
Function
Rule that assigns a single value y to each x.
Vertical Line Test
No vertical line intersects the graph more than once.
Even Function
f(-x)=f(x)
Odd Function
f(-x)=-f(x)
Composite Function
f of g(x) = f(g(x))
Exponential Fucntion
y=a^x where a>0 and x is a variable.
Natural Exponential Function
When a=e.
Logarithimic Function
log(base a)x is defined as the inverse of the exponential function y=a^x.
Piecewise Function
Uses different formulas on different parts of its domain.
Vertical Asymptote
If the function approaches positive or negative infinity as x approaches a from either the left or right.
Function is continuous if:
If the left limit equals the right limit, and if f(a)=L.
Function is differentiable if:
The derivative exists there.
sin(pi/6)
1/2
sin(pi/4)
sqrt(2)/2
sin(pi/3)
sqrt(3)/2
cos(pi/6)
sqrt(3)/2
cos(pi/4)
sqrt(2)/2
cos(pi/3)
1/2
tan(pi/6)
sqrt(3)/3
tan(pi/4)
1
tan(pi/3)
sqrt(3)
sin(0)
0
sin(pi/2)
1
sin(pi)
0
sin(3pi/2)
-1
sin(2pi)
0
cos(0)
1
cos(pi/2)
0
cos(pi)
-1
cos(3pi/2)
0
cos(2pi)
1
