Impossible Calculus - Not For The Faint of Heart! Quiz / Test | Sci / Tech

Quiz Answer Key and Fun Facts

1. As a follow-up of the instructions, no Fill-In-The-Blank question will have any special characters in the answer. Let's begin! There is a new population of deer in a field in a National Park. Their growth can be modeled logistically (not exponentially - after all, there is only so much food and living space) based on the formula (deer vs. time): d(t)=(1500e^t-20)/(1000+20e^t-20). What is that field's carrying capacity? (The maximum number of deer it can sustain?)

Answer: 75

Using L'H�pital's Rule (because as t→∞, both the numerator and the denominator do as well), take the derivative of the top and the bottom separately to obtain: (1500e^(t-20))/(20e^(t-20)). Now, to compute the limit as t→∞ you can simply cancel out the e^(t-20) terms to obtain 75. Hooray for practical uses of L'H�pital's Rule!

2. A car travels along the path such that the displacement vs. time (meters vs. seconds) graph is given by the equation: x(t)=(6/π)sin^-1(t)*cos^-1(t). What is the instantaneous velocity of the car at the point (1/2, π/3)?

Answer: 2/√3 m/s

The instantaneous velocity of the car is dx/dt, or just the derivative of the function evaluated at the given point. First, use the product rule to evaluate the derivative. The derivative of this equation is (6/π)(cos ̄�t-sin ̄�t)/√(1-x�). The derivative evaluated at the point (1/2, π/3) is 2/√3 m/s, which is your instantaneous velocity.

As an aside, if you wanted to find the equation of the tangent line, you can then set up the equation (in the form x=mt+b): π/3=(2/√3)(1/2)+b. Solving for b yields b=(π-√3)/3 so the equation of the tangent line is
x=(2/√3)t+(π-√3)/3.

Note: The symbols for the inverse trig functions might look funny depending on your computer.

3. Annie has a 2-inch piece of chalk. Each inch of chalk can draw 1000 feet of lines. She wants to draw a curve modeled by the equation x^(3/2). She envisions the curve drawn on a plot where each unit on the x-axis and y-axis is equal to 1 yard (1 yard=3 feet). She draws the curve starting at the origin and ending at the point (44, 44^(3/2)). How many inches of chalk does Annie have remaining after she draws this curve?

Answer: 1.112

The derivative of this function is (3/2)√x. Putting this into the arc length formula arc length = ∫√(1+f'(x)�)dx and evaluating the integral from x=0 to x=44 leaves you with (8/27)(1+9x/4)^(3/2). Plugging in 44, plugging in 0 and subtracting leaves you with 296 YARDS. Multiply 296 by 3 to get 888 feet. (888 feet)/(2000 feet-per-piece) leaves 44.4% of the chalk used which means she has 56.6% � 2 inches = 1.112 inches left.

4. Sylvia the potter has just completed a large order of ornate clay bowls but has just realized that she is one short. She only has 1 cubic foot of clay left. Does she have enough to make the last bowl? (The bowl can be modeled by region bounded by y=x^2 and y=√x when rotated around the x-axis.)

Answer: Yes, with about 0.05 cubic feet to spare

The formula for rotation about an axis is π∫(R(x)�-r(x)�)dx. In this case these functions intersect at (1,1) so the bounds of the integral are [0,1]. The R(x) function is √x and the r(x) function is x�. Evaluate the integral
π∫(x-x⁴)dx over the interval [0,1] and you'll get 3π/10. 3π/10 approximates to 0.942 so she'll have enough clay for the bowl with about 0.058 cubic feet to spare.

5. A strange and unusual particle moves with velocity equal to v(t)=t^10*ln(t) m/s. You observe this particle carefully in a lab and find that over the time interval [1, e^(1/11)] the particle moves _______ meters. Hint: Integration by parts is used to find the antiderivative of a product. Its formula is ∫udv = uv-∫vdu.

Answer: 1/121

Distance is simply the integral of velocity.

Let lnt=u and (t^10)dt=dv. Then du=(1/t)dt and v=(t^11)/11. So, by the formula in the question the integral becomes: lnt*(t^11)/11 - ∫((1/t)*(t^11)/11)dt. Note the cancellation that will occur in the integrand. Evaluating this integral like you would any other leaves you with: lnt*(t^11)/11 - (t^11)/121 or (t^11)*(11*lnt-1)/121. Evaluating this with the bounds given in the question leaves you without any e terms and simply 1/121.

6. An old man with a bad back wants to hang a plane mirror on his wall and wants to know what angle it will make with the floor so that he can lie down on his back under it and watch TV from across the room. The problem is, his house was designed by Frank Gehry and the wall is modeled by the equation z=�x^2+�y^2. He wants to attach the mirror to the wall tangent to the point (2,4,6). What angle will this tangent mirror (plane) make with the floor? (You can assume that the floor is represented by the equation z=0.)

Answer: 70.5�

This starts like a tangent plane problem. Find the partial derivatives with respect to x (p.d.x.) and y (p.d.y.) and evaluate them at the point (2,4,6). The p.d.x. evaluated at (2,4,6) is 2 and the p.d.y.evaluated at (2,4,6) is 2 as well. Plugging these values into the formula for tangent planes --- z-z₀=fx(x-x₀)+fy(y-y₀), where fx is the p.d.x. evaluated at the given point and fy is the p.d.y. evaluated at the given point --- yields: z-6=2(x-2)+2(y-4). Simplifying leaves you with: 2x+2y-z=6.

To find the angle between this plane and the plane z=0 is a matter of finding the angles between their normal vectors, [-2,-2,1] and [0,0,1], with lengths 3 and 1, respectively. We can use the dot-product here. u*v=|u||v|cosθ. Evaluating this with these two vectors gives us the equation 1=3cosθ. Solving for the angle θ yields θ=70.5�

7. A factory producing computers has two factors that contribute to production: labor (L) and capital (K). The number of computers they can produce in an hour is given by the function P=100L^0.4K^0.6. Unfortunately, the factory's budget is $3000 per hour and one unit of labor costs $75 per hour and one unit of capital costs $100 per hour. To produce the most computers in one hour, how many units of labor should the factory employ and how many units of capital should they use? Write your answer as the ordered pair L,K where L is the number of units of labor and K is the number of units of capital.

Answer: 16,18

Using the Method of Lagrange Multipliers, the gradient of the production function equals some scalar times the gradient of the constraint: [[40L^(-0.6)*K^(0.6),60L^(0.4)*K^(-0.4)]] = [[75λ,100λ]]. (The brackets indicate vectors.) Set up three equations from this:
a. 40L^(-0.6)*K^(0.6)=75λ
b. 60L^(0.4)*K^(-0.4)=100λ
c. 75L+100K=3000

Dividing the first two equations (b/a) yields: 3L/2K = 4/3 or K = 9L/8.
Plugging into the third equation (c) yields: 75L+900L/8 = 3000 or L=16.
Using K = 9L/8, we find that K = 18.

Solving this system of equations yielded L=16 and K=18. (This lets them produce approximately 1717 computers per hour. Not bad.)

Thanks to Diceazed for catching an algebraic error I had here.

8. Everyone (OK, some) people know that e^(x^2) is not integrable by normal means of antidifferentiation. However, in double integrals it is possible sometimes to reverse the order of integration (do dy dx instead of dx dy) in order to solve an otherwise impossible integral. Try solving this: ∫[0;1] ∫[3y;3] e^(x^2)dxdy (The bounds are [0,1] and [3y,3] in case they're hard to read there.)

Answer: (e⁹-1)/6

You can reverse the order of integration by changing the bounds and swapping the dx and the dy. The current area of integration is a right triangle with corners (0,0);(3,0);(3,1). The line through the origin and (3,1) is the line x=3y and the line through (3,0) and (3,1) is the (vertical) line x=1. Instead of saying that x goes from 3y to 3, you can say that y goes from 0 to x/3 (rewriting the equation x=3y). You can then say that x goes from 0 to 3. So your new integral is: ∫∫e^(x^2)dydx with the y bounds being [0,x/3] and the x bounds being [0,3]. Note the placement of the dy and the dx. You can now solve this integral normally. After integrating one time, you're left with: ∫₀�(x*e^(x^2)/3)dx. You can use u-substitution to solve this integral and obtain (e⁹-1)/6.

Sorry about the format. HTML is not allowed in the solutions so I can't do as many subscripts and superscripts.

9. You go out for dinner at Papa Raboloid's Italian Bistro and order his signature 16π drink. He pours 16π cubic centimeters of reddish liquid into your paraboloidic cup. (Your cup can be modeled by the equation z=2x^2+2y^2.) Your cup is 10 cm tall. How many centimeters will be left unfilled at the top of the cup? Hint 1: A paraboloid is a 3D parabola with a circular top, and the most basic form is given by z=x^2+y^2. Hint 2: It may be helpful to use polar coordinates.

Answer: 2

Start by converting the integral to polar coordinates.∫∫(2x�+2y�)*dydx = ∫∫2r�*rdrdθ. The equation z=2x�+2y� can also be expressed as z=2r�, because x�+y�=r�. We want to find z, but in order to perform the integration, we need r in terms of z. r=√(z/2). If the volume of the liquid is 16π, we can set up the integral we found before. ∫∫2r�*rdrdθ=16π with the r bounds being [0,√(z/2)] and the θ bounds being [0,2π]. Solving this integral equation yields the equation: π*z�/4=16π. Canceling the π's, multiplying by 4, and taking the square root, you find that z=8. Since the cup is 10 cm tall, and the liquid fills 8 cm, there are 2 cm left unfilled at the top of the cup.

10. Luke Skywalker travels in his X-Wing along a path C given by the boundary of the area enclosed by a semicircle of radius 3 units, semicircle of radius 1 unit, and the x-axis. (This path is entirely in Quadrants I & II.) A variable force field, created by Darth Vader and given by (2y^2)i+(8xy)j --- (i and j are the unit vectors [[1,0]] and [[0,1]]) --- is present. How much work is done on Luke by Vader's force field as he travels completely around the closed loop once? Write your answer as an improper fraction a/b without any units.

Answer: 208/3

This is Green's Theorem. The full path, starting at the point (1,0) is:
a. (1,0) to (3,0) along the x-axis.
b. (3,0) to (-3,0) along the polar curve r=3.
c. (-3,0) to (-1,0) along the x-axis.
d. (-1,0) to (1,0) along the polar curve r=1.

Green's theorem states that ∫(Pdx+Qdy) around a closed loop is equal to: ∫∫(dQ/dx-dP/dy)dA. In this case, P is the i-vector force, and Q is the j-vector force. It follows that dQ/dx=8y and dP/dy=4y, so dQ/dx-dP/dy=4y*dA. Since we're working in polar, 4y*dA=4rsinθ*rdrdθ. From here you can do an iterated integral: ∫∫4r�sinθ*drdθ with the r bounds being [1,3] and the θ bounds being [0,π]. (It's only 0 to π because it's a semicircle.) Upon evaluating the double integral, you get 208/3 as your final answer. May the force be with you.

I hope you enjoyed the quiz.

Source: Author redsoxfan325

This quiz was reviewed by FunTrivia editor crisw before going online.
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Impossible Calculus - Not For The Faint of Heart! Quiz

This quiz is designed to test your knowledge of topics covered in Calculus I through Calculus III. Don't attempt this unless you know this material well or you'll find it very difficult and probably score 0/10. Good luck and may the best mathlete win!

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