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Chapter 5
Random Trig
100
Suppose the equation of a line is y = 2x + 4. What is the slope of a line perpendicular to this line?
-1/2
100
Recall that a rational function is the quotient of two polynomials. (e.g. f(x) = g(x) / h(x)). How does one find the zeros of a rational function?
The zeros of f(x) are the zeros of g(x) that are not zeros of h(x)
100
Use the sum formula for cosine to expand: cos(x + y)
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
100
How is the graph of f(x) = -e^{x-2} + 1 different from the graph of f(x) = e^x?
Shifted right 2 and up 1 and flipped over the x-axis
100
sin(pi / 4)
1/sqrt(2) = sqrt(2)/2
200
Find the equation of a line in point-slope form that passes through the points (2,4) and (3,9).
y = 5(x - 2) + 4 OR y = 5(x - 3) + 9
200
How does one find the holes of a rational function f(x) = g(x)/h(x)?
Holes are the shared zeros of g(x) and h(x)
200
Use the sum for sine to expand: sin(x + y)
sin(x + y) = sin(x)cos(y) + sin(y)cos(x)
200
Break this down as much as possible: ln[ (x^2 * sqrt(y)) / ( (z+2)^3 ) ]
2ln(x) + 1/2ln(y) - 3ln(z+2)
200
cos(5 pi / 6)
- sqrt(3)/2
300
Find the inverse of the function f(x) = 8x + 2.
f^{-1}(x) = (x-2)/8 = x/8 - 1/4
300
How does one find the vertical asymptotes of f(x) = g(x)/h(x)?
Vertical asymptotes are the zeros of h(x) that are not also zeros of g(x)
300
Write down the law of cosine
a^2 = b^2 + c^2 - 2bc cos(alpha)
300
Put this back together: (1/2)ln(x) + 4ln(y) - 8ln(z)
ln [ (x^{1/2}y^(4) ) / z^8 ]
300
If cos t = 1/2 then t =
pi / 3 OR 5*pi/3
400
Write down the difference quotient
f(x+h) - f(x) / h
400
Let f(x) = g(x)/h(x), and let the degree of g(x) = m and the degree of h(x) = n. Describe how to find horizontal asymptotes given (1) m = n, (2) m > n, (3) m < n.
m = n => divide leading coefficients m > n => no horizontal asymptote m < n => x-axis is a horizontal asymptote
400
Write down the law of sine
sin(A)/a = sin(B)/b = sin(C)/c
400
A substance decays by 40% in 10 yrs. Write down an equation that you can use to find the decay rate k.
.60 * Q0 = Q0 * e^{10*k}
400
arcsin ( sin (1/2) )
1/2
500
How does one use the difference quotient to find the instantaneous rate of change of a function?
Evaluate the difference quotient at h = 0
500
Let f(x) = (4(x-5)(x+1)^2)/(x+3)(x-5). Find the zeros of f(x) and state their multiplicity.
x = -1, multiplicity 2
500
Find all values of x in the interval [0,2pi] that satisfy the equation (3 cot x)^2 = 27
pi/6, 5pi/6, 7pi/6, 11pi/6
500
Suppose you know the decay rate k for a substance. Write down an equation you can solve to find the half life of the substance.
t = ln(.5) / k
500
cos(arcsin(1/2))
sqrt(3)/2