Limits
Derivatives
Critical Points
Graphs
Exponentials
100
Find
lim_(x->∞)(-x²+4x-5)
-∞
100
d/dx((7x)(x+3))
14x+21
100
Find the critical point of the function:
f(x)=x²+x
x=-1/2
100
lim_(x->0^-)(g(x))
-2
100
What function is the inverse of
f(x)=ln(x)?
f(x)=e^x
200
Find
lim_(x->∞)(3x^3-5x)/(x^3+6x²-4
3
200
d/dx(g(f(x))
g'(f(x))f'(x)
200
Find the critical points of f(x).
f(x)=1/3x^3+x²-3x
x=-3, x=1
200
f'(x) is undefined when?
x=3
200
What is the formula for the Effective Annual Rate (EAR) when interest is compounded continuously?
EAR = e^r-1
300
Write the definition, in terms of a limit as h -> 0, of the derivative of a function f(x).
f'(x)=lim_(h->0)((f(x+h)-f(x))/h)
300
d/dx(e^x/(4x))
(xe^x-e^x)/(4x²)
300
Find the point of inflection of f(x).
f(x)=1/3x^3+x²-3x
x=-1
300
Is f''(2) > f''(0)?
Yes - f(x) is concave up at x=2 and concave down at x=0!
300
Compute
H'(x)
where
H(x)=x^7ln(x)
H'(x)=x^6(7ln(x)+1)
400
Find
lim_(x->2)((x²-4)/(x-2))
2
400
d/dx(3(e^x-2)^4))
12(e^x-2)^3(e^x)
400
f(x) has a critical point at x = -1. Is this a local minimum, a global minimum, a local maximum, or a global maximum?
f(x)=3x²+6x-4
Global Minimum
400
Find J'(4) where J(x) = f(x)g(x)
13
400
Find all real solutions to the equation:
log_10(x+1)-log_10(2x)=1
x=1/19
500
Does f(x) have a horizontal asymptote? If so, what is it?
f(x)=(x²+42x+645)/(55x^3-1029x²-3x)
Yes, at y=0
500
d/dx(ln((4x²-6x)^3))
(3(8x-6))/(4x²-6x)
500
Find the interval(s) on which f(x) is concave down.
f(x)=x^4/4-x^2/2
[-1,1]
500
q=g(g(x))
q'(4) = ?
9
500
What was the initial investment if an account accrues 3% annual interest, compounded monthly, and grows to $90,000 after 10 years?
$66,698.61