Week One
What is the domain of ln(x^2-c^2)?
(-\infty, -c) \cup (c, \infty)
Give an example of a non-positive function with a corner at x=c.
-|x-c|
Find the fifth derivative of the function e^{-2x}.
-32 e^{-2x}
An expanding sphere’s radius grows at 2 cm/s. How fast does its volume grow when the radius is 5 cm?
200 \pi cm^3/s
Give an example of a cubic function with no critical points.
x^3 + x
Give an example of a continuous rational function with a real domain that is not a polynomial.
\frac{1}{x^2+1}
Show that |x| is not differentiable at x=0 using the limit definition of the derivative.
\lim_{h \to 0^+} \frac{|h|}{h} = 1 AND \lim_{h \to 0^-} \frac{|h|}{h} = -1
Find the fifth derivative of the function ln(x+1).
\frac{24}{(x+1)^5}
Find two positive numbers whose sum is 67 and whose product is as large as possible.
33.5 and 33.5
Give an example of a decreasing, concave down function with a real domain.
-e^x
Give an example of a non-negative quartic (degree 4) polynomial function with two roots a and b.
(x-a)^2 (x-b)^2
Give an example of a power function that has a cusp at x=c where x>=c.
\sqrt{x-c}
Find the fifth derivative of the function sqrt{x}.
\frac{105}{32} x^{-\frac{9}{2}}
An open-topped rectangular box with a square base is made from 192 square feet of material. What is the side length of the base that maximizes the volume of the box?
8 ft
Give an example of an increasing, concave down function with a real domain.
-e^{-x}
Give an example of a non-negative rational function that has two vertical asymptotes x=a and x=b.
\frac{1}{(x-a)^2(x-b)^2}
Give an example of a power function that has a vertical tangent line at x=c.
(x-c)^\frac{1}{3}
Find the fifth derivative of the function sin(2x+1).
32 \cos{2x+1}
A Needoh ® is thrown vertically off of a platform that is 5 feet above the ground at a velocity of 17 m/s. Given that the acceleration due to gravity is 9.8 m^2/s, determine the amount of time (in seconds) it took for the Needoh to reach its maximal height.
1.7347 s
What is the indefinite integral of e^{2x}?
e^{2x}/2 + C
Give an example of a rational function that has one root x=a, one removable discontinuity x=b, and one vertical discontinuity x=c.
\frac{(x-a)(x-b)}{(x-b)(x-c)}
Show that x e^x = 1 has a solution using the Intermediate Value Theorem.
f(x) = x e^x is continuous on [0,1] and f(0) = 0 and f(1) = e. Since 1 is between 0 and e, there exists c between 0 and 1 such that f(c)=1.
Find the fifth derivative of the function x * e^{x}.
(x+5) e^x
Find the global maximum of the xth root of x where x is positive.
e^\frac{1}{e}
What is the indefinite integral of 6x/sqrt{4x}?
2x^\frac{3}{2}+C