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Rational Functions
Exponential Functions
Log Rules
Log/Exponential Applications
Exam 1 Review
100
What are the 6 steps to graphing a rational function?
1: find domain 2: reduce to lowest terms 3: find x and y intercepts 4: find vertical asymptotes and holes (if any) 5: find horizontal asymptotes 6: sign diagram/analyze behavior
100
f(x) = 4^(3x) solve for f(1/6)
f(1/6) = 2
100
How do you write natural log and common log?
ln(x) log (x) 10
100
Find the amount of interest paid on an investment of $2400 after 5 years at 7%.
$840
100
A line of slope 34 passes through the point (2000, 12345). Find the equation for the line in POINT-SLOPE form
y-12345 = 34(x-2000)
200
find horizontal asymptote of: x^2-1 ----------- 3x^2+x-4
y= 1/3
200
solve: 5^(3x) = 5^(7x-2)
x = 1/2
200
Evaluate: log (216) 36
3/2
200
Find the decay constant k of Chromium 51, used to track red blood cells, initial amount 75 milligrams, half-life 27.7 days. *round to 4 decimal places
k ≈ −0.0250
200
Find the zeros of the polynomial: x^2 +16
x= +4i x= -4i
300
Find the x and y intercepts of: x^2-1 ----------- x^2+x-6
x-int: (-1,0) (1,0) y-int: (0, 1/6)
300
What is the inverse of an exponential function?
f^-1(x) = log (x) b
300
What are the 3 rules and 3 properties of logarithms?
Rules: 1) log (MN) = log (M) + log (N) a a a 2) log (M/N) = log (M) - log (N) a a a 3) log (M^r) = rlog (M) a a Properties: 1) log (x) = y <--> b^y = x b 2) log (b^x) = x b 3) b^(logb(x)) = x , x> 0
300
The diameter D of a tumor, in millimeters, t days after it is detected is given by: D(t) = 15e^(0.0277t) How long until the diameter of the tumor doubles?
approx. 25 days
300
put into standard form: f(x)=2x^2-8x+9
2(x-2)^2 + 1
400
Find any vertical/horizontal asymptotes and holes of: 4x^2-36 -------------- x^2-5x+6
V.A.: x=2 H.A.: y=4 Hole: (3, 24)
400
Given the original function f(x) = 2^x, describe the transformation of: g(x) = -2^(x+3)+4
left 3 units reflection about the x-axis up 4 units
400
solve: log (3x+7) - log (5x-9) = 1/2 169 169 *round to 4 decimal places
x=2
400
List the formulas for: 1: simple interest 2: compound interest 3: continuously compounding interest 4: exponential growth 5: exponential/radioactive decay
1: A = P(1+rt) 2: A(t) = P(1+r/n)^(nt) 3: A = Pe^(rt) 4: N(t) = N e^(kt) where k > 0 0 5: A(t) = A e^(kt) where k < 0 0
400
Suppose f(x) = e^(4x+12) and g(x) = 1/4ln(x)-3 compute (fog)(x) and (gof)(x) BONUS 100 = explain the reason behind your answers
(fog)(x) = x (gof)(x) = x
500
Find: 1: domain 2: x and y intercepts 3: vertical asymptotes, horizontal asymptotes, and holes (if any) 4: sign diagram and graph 2x^2-9 ----------- x^2-9
1: (−∞, -3)U(-3,3)U(3,∞) 2: x-int: (-1,0) (1,0) y-int: (0, 1) 3: V.A.: x=-3 x=3 H.A.: y=9 no holes 4: +(-3)-(-1)+(0)+(1)-(3)+
500
complete table and graph f(x) = 8^(1-x) x | y -2 | 0 | 1 | 3 | | 8 | 16 | 32 | 64 | 1/8 | 1/32 | 1/64
512 8 1 1/64 0 -1/3 -2/3 -1 2 8/3 3
500
expand log and simplify: log [27^6(3x)^4]/[9^8] 3
6+4log (x) 3
500
Suppose that I borrow $1000 at an annual interest rate of 4%. How much do I owe after 3 years if... 1: the interest is compounded annually 2: the interest is compounded quarterly 3: the interest is compounded monthly 4: the interest is compounded daily 5: the interest is compounded continuously
1: $1124.86 2: $1126.83 3: $1127.27 4: $1127.49 5: $1127.50
500
g(x) = 4(x+6)^2(x-3)(x^2+16) 1: describe the end behavior 2: find all real zeros 3: find all complex zeros 4: find multiplicity of real zeros
1: as x -> ∞, f(x) -> ∞ as x -> -∞, f(x) -> -∞ 2: x = -6, 3 3: x = +4i, -4i 4: x=-6 mult. of 2 x=3 mult. of 1