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52 Cards in this Set
- Front
- Back
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function
(memorize) |
a function F is a rule that assigns to each x a value f(x)
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Domain
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the set of all values for which the rule makes sense
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vertical line test
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a curve in the xy plane is the graph of a function if and only if no vertical line intersects the curve more than once
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what functions are continuous?
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linear (y= x+1)
polynomials (y= 3x^2 +5x +1) power (y = x^3) |
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y = f(x) + c
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shift upward by c units
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y = f(x) -c
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shift downward by C units
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y = f (x+c)
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shift to the left c units
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y = f (x-c)
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shift to the right c units
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y = cf(x)
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stretches the graph vertically by a factor of c
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y = 1/c f(x)
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compresses the graph by a factor of c vertically
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y = f (cx)
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compresses the graph horizontally by a factor of c
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y = f(x/c)
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stretches the graph horizontally by a factor of c
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y = -f(x)
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reflection about the x axis
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y = f(-x)
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reflection about the y axis
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degrees and radians
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pi = 180 degrees
pi/2 = 90 pi = 180 3pi/2 = 270 2pi = 360 |
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sin (angle)
cosine (angle) |
sin: y coordinate
cosine: x coordinate |
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sin^2 + cos^2 =
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sin^2 + cos^2 = 1
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one-to-one function
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A function is called one-to-one if it takes no value twice
(does it pass the horizontal line test?) |
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How do you find the inverse of a function?
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Solve the function in terms of x, then switch x and y!
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Logs
y = log(sub a)X equals... |
a^y = x
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log (e) X =
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ln (x)
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Sin is not one-to-one, so what range and domain do we use?
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range: [-1,1]
domain: [-pi/1, pi/1] |
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Velocity formula
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S(t2) - S(t1)
____________ t2-t1 |
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Limit does not exist if...
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there is division by 0
limits of (-) and (+) sides do not equal the same value |
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direct substitution property
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if f(x) is a polnomial or a rational function, and a is in the domain of f then lim (x--> a)(F(x)) = f (a)
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Squeeze Therm
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Fx < Gx < Hx for x near a (except possibly at a) and lim x --> a Fx = L = lim x--> a h(x)
-1 < sin 1/x < 1 for all x near 0 -IxI < IxI sine 1/x < IxI for all x near 0 |
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Continuity
(memorize!!!) |
A function f is continuous at a number a if lim (x --> a) = F (a)
A function is continuous on an interval if it is continuous at every point in that interval. |
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What kinds of functions are continuous?
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polynomials, rational, trig, inverse, exponential, logs, roots.
FUNCTIONS FOR WHICH THE DIRECT SUBSTITUTION PROPERTY HOLDS!! |
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Intermediate value property
(memorize!!) |
f is continuous on [a,b] and N is any number between F(a) and F(b) then there is a number c in (a,b) such that f(c) = N
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Slope of the tangent line of F at A is...
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M= lim (x --> a) [ f(x)-f(a)/(x-a)]
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Derivative formula...
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f'(a) = lim (x --> a) [ f(x)-f(a)/(x-a)]
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What is a derivative?
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The slope of the tangent line to the graph
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Other derivative formula- "find the slope of y = blah at a = blah"
**Same as instantaneous velocity MEMORIZE |
f(x) = lim (h--> o)
[f(a + h) -f(a)]/h where h = x-a x = a+h |
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cosine
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domain: all real #s
range: -1,1 |
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sine
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domain: all real #s
range: -1,1 |
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tangent
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Domain: all real but 0
range: all real #s |
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cotangent
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Domain: all real but 0
range: all real #s |
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cosecant
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Domain: all real but 0
range: -1,1 |
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secant
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Domain: all real but 0
range: -1,1 |
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f^-1(x) =
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[f(x)]^-1
NOT 1/f(x) |
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Direct substitution property
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if F is a polynomial or rational and a is in the domain of F, then lim (x -> a) (f(x) = f(a).
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lim (x--> a) f(x) = L if an only if
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limit from (-) side = limit (+) side
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Squeeze Theorm
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If f(x) < g(x) < h(x) when x is near a (except possibly at a and lim f(x) = lim h(x) = L then lim g(x) = L
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Continuity
MEMORIZE |
A function f is continuous at a number a if lim (x-> a) f(x) = f(a). A function f is continuous on an interval if it is continuous at every number in the interval.
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Derivative as a function formule
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lim (h -> 0) [f(x+h) - f(x)]/h
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Limit Law: sum law
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the limit of a sum is the sum of its limits
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Limit law: difference law
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the limit of a difference is the difference of the limits
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limit law: constant multiple law
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the limit of a constant times a function is the constant times the limit of the function.
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limit law: product law
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the limit of a product is the product of the limits
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quotient law
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the limit of a quotient is the quotient of the limits
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What is a limit?
memorize |
Lim (x--> a) f(x) = limit if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a, on either side of a, but not equal to a.
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What is a derivative?
memorize |
The derivative f'(a) is the instantaneous rate of change if y= f(x) with respect to x when x = a.
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