Rotational Solids
Which is Bigger?
Optimization
Integration
Rules & Theorems
100
It's the volume of the solid generated by rotating the region formed by y = x, the x-axis and x = 2 about the x-axis.
What is 8pi/3 ?
100
It's the bigger value between 1 and...
lim_(h\rightarrow0)(sin(pi/2+h)-sin(pi/2))/h
What is 1 ? (Limit = 0
100
If a piece of cardboard that is 14 inches by 10 inches is cut along its corners by equal lengths to be folded up in a box, it's the cut length that will maximize area. (Calculator OK)
What is 1.9183 inches cuts?
100
int_0^pi(3x^2+sinx)dx
What is
pi^3+2
100
It's true or false: A function f(x) can have an interval in which the Extreme Value Theorem is applicable but not the Mean Value Theorem.
What is true? (f is not differentiable but continuous)
200
It's the volume of the function f(x) rotated about the x-axis.
f(x)=2sqrtx
What is 2pi?
200
It's the larger of ...
lim_(x\rightarrow\infty)(e^x/x^e) , lim_(x\rightarrow\infty)(x^e/e^x)
What is...
lim_(x\rightarrow\infty)(e^x/x^e)
200
A rectangle is inscribed inside a circle of radius 3 with length l and width w. It's the dimensions of the rectangle that maximizes the quantity lw^2
What is...
l=2sqrt3 , w =2sqrt6
200
It's the indefinite integral of...
int(1/(xsqrt(x)))dx
What is...
-2/sqrtx+C
200
It's the proof that f'(0) does not exist for f(x)=|x| using limits.
What is...
lim_(h\rightarrow0^+)(f(x+h)-f(x))/h \ne lim_(h\rightarrow0^-)(f(x+h)-f(x))/h
300
It's the integral for the volume of the solid formed by rotating the region bound by y = cos x, y = 0.5 and the y-axis about the x-axis.
What is...
piint_0^(pi/3)(cos^2(x)-0.25)dx
300
A function f is positive on the interval [1, 5]. Given that f is decreasing with a positive second derivative, it's the larger of a left Riemann sum, midpoint Riemann sum or right Riemann sum approximating...
int_1^5f(x)dx
What is the left Riemann sum?
300
It's the point on the parabola y=x^2 that's closest to the point (3, 0)
What is (3/2, 9/4)
300
It's the indefinite integral of...
int(45/(27+75x^2))dx
What is...
arctan(5/3x)+C
300
A function f(x) is differentiable on (a, b). It's the value of f'(x) that must exist if f(a)=f(b)
What is f'(c) = 0 ?
400
It's the integral representing volume of the solid generated when the function f(x) is rotated about the line x = 2...
(x+4)^2+(y-4)^2=4
What is...
V=piint_2^6((-sqrt(4-(4-y)^2)-2)^2-(sqrt(4-(4-y)^2-2))dy
400
A cup has the shape of a cone with height of 12 cm and a radius of 4 cm at the top. Coffee is being poured into the cup at a rate of 2h cm^3/sec, where h is the height in cm. It's the level at which the coffee level rising is greater, 6 cm or 3 cm.
What is 3 cm ? (3/pi @ 6 vs 6 / pi @ 3)
400
A hipster wants to build a rectangular room in their house with one wall being exposed brick and the rest being homemade wallpaper. The brick costs $50/foot while the other 3 sides of wallpaper cost $15/foot. With a budget of $1050, it's one of the dimensions that the room should be to maximize area?
What is a length of 105/13 feet and a width of 15/2 feet?
400
It's the indefinite integral of...
int(3/(9x^2+1))dx
What is...
arctan(3x)+C
400
It's the maximum and minimum number of points of inflection a polynomial of odd degree n can have.
What is a maximum of (n - 2) and a minimum of 1?
500
It's the volume of the solid formed by rotating the region bound by y = cos(x^2), the x-axis and x=1 about the line x = 1 pictured below:
What is...
pisin(1)
500
A type of bacteria grows at a rate dy/dt = ky , where y is the population and t is the number of years. If one population triples every 7 years and another doubles every 5 years, it's the population that will be bigger after 12 years.
What is the population tripling every 7 years?
500
A pyramid has a volume of 20 ft^3 with a square base and is formed from a square and 4 isoceles triangles. The square costs $1 per square food and the triangles cost $16/15 per square foot to create. It's the dimensions of the pyramid that minimizes the cost.
What is width of 4 and height of 3.75 feet?
500
It's the infinite limit summation equivalent to...
int_3^5x^4dx
What is...
lim_(n\rightarrow\infty)\sum_(k=1)^n2/n*(3+(2k)/n)^4
500
Given the following inequality below which is true on the interval [-1, 1], determine the given limit WITHOUT use of L'Hospital's Rule.
cos(x)<=sin(x)/x<=sec(x) , lim_(x\rightarrow0)(sinx/x)
What is the use of the Squeeze Theorem (since both upper and lower functions have limits at 0 going to 1)?