Calculus Test Solutions

Calculus Pretest: Solutions May 10, 2009

1

Compute 1.

d 3 dx 5x

− 4x2 − 3x + 1 = 15x2 − 8x − 3

2.

d dx

4.

d dx

ln(x) =

5.

d dx

log5 (x) =

6.

d dx f (4x)

7.

d dx

R x2

8.

d dx

R x2

9.

d dx f

sin(x) = cos(x)
d 3. dx tan x1 = x2 cos−1 2 (1/x)

0

0

1 x 1 ln(5)x

= 4f 0 (4x)

f (t)dt = 2xf (x2 ) ln(x)f (t)dt = 2x ln(x)f (x2 ) +

d 4 3 dr 3 πr

11.

d 2 dr πr

12.

d i(kx−ωt) dx e

13.

d e−x dx x

14.

d dx

15.

d ex +e−x dx 2

=

ex −e−x 2

16.

d ex −e−x dx 2

=

ex +e−x 2

= 2πr = ikei(kx−ωt)
−x = −ex 1 − x1 1 1+x2

sin(mx) sin(nx)dx = n; π − 14 sin(4πn), n = m 0

1 x f (t)dt

= 4πr2

arctan(x) =

R 2π

0

(t(x)) = f 0 (t(x)) t0 (x)

10.

17.

R x2

−n n2 −m2

m cos(2πn) sin(2πm)+ n2 −m 2 sin(2πn) cos(2πm), m 6=

1

sin(x)ex dx =

18.

R

19.

R1

20.

R∞

21.

R∞

22.

R∞

23.

R∞

0

ex (sin(x)−cos(x)) 2

x3 cos(x)dx = −3 sin(1) + 5 cos(1)

0 0 0

e−αx dx =

1 α

xe−αx dx =

1 α2

xn e−x dx = n!

−∞

2

x2 e−x dx =

√ π 2 .

24. ıı = e−π/2+2πk , k = 0, ±1, ±2, . . . 25. The local minima and maxima of x3 e−x are a maximum at x = 3. 26. The first three terms in the Taylor series expansion of tan(x) about x0 = are tan(x) ≈ 1 + 2x + 2x2 .

2

π 4

Interpret 1. You’re hiking down a trail. What is the derivative of your position with respect to time? (By “what is” I mean “give an interpretation for” not “what is the numerical value?”) Answer: your hiking speed 2. You’re hiking up a hill. What is the derivative of elevation with respect to position? Answer: The slope of the hillside. 3. You’re still hiking up that hill. Your GPS measures your position and altitude at all times. How can you use this data to find the slope of the hill? Answer: The derivative of your altitude with respect to time divided by the derivative of your position with respect to time gives the slope of the hill. 4. What is the derivative of the energy needed to compress a gas with respect to the volume it occupies? Answer: pressure 5. What is the derivative of distance driven down a road with respect gas in your tank? Answer: gas mileage 6. What is the derivative of money paid at the pump with respect to gas in your tank? Answer: per-gallon price 7. What is the derivative of gas in your tank with respect to distance driven? Answer: gallons per mile, the reciprocal of miles per gallon. 8. What is the derivative of potential energy with respect to position? Answer: force

2

9. What is the derivative of temperature with respect to heat? Answer: heat capacity 10. What is the derivative of total mass with respect to volume considered? Answer: density 11. What is the derivative of distance you’ve driven down the road with respect to how many times the wheels have revolved? Answer: the diameter of the wheel 12. What is the derivative of the height of the ocean with respect to volume of water that melts off continental ice shelves and into the ocean? Answer: the reciprocol of the surface area of the ocean 13. A shark smells blood in the water and is trying to get to it. What does this have to do with derivatives? Answer: The shark should swim in the direction that maximizes the derivative of the density of blood. 14. At the end of a 200m race, is the derivative of a runner’s position (distance from start line) positive or negative? Positive. The runner is getting further away from the start line. For a 400m race, the answer is reversed, because the runner is approaching the start line having completed nearly one lap. 15. What about the second derivative of position? Answer: A tricky question, because the runner is slowing down, which would make the second derivative negative, but the runner is also moving more directly away from the start line due to the curve of the track, which would make the second derivative positive. Using some reasonable dimensions for an outdoor track, I find the second derivative is positive whenever v 2 /1m > 150dv/dt with v the magnitude of the runner’s velocity and 1m one meter. This condition should be met unless the runner is “pulling up” early. 16. When parachuting, which of the following are positive and which negative: height above ground, first derivative of that height, second derivative of height, third derivative of height, fourth derivative of height? Answer: Height above ground is positive. You’re falling, so the first derivative with respect to time is negative. Gravity accelerates you down, so the second derivative is also negative. Wind resistance pushes you back up, and the wind resistance increases with time because you’re going faster. That makes the third derivative positive. Your speed reaches an asymptote, so the rate at which wind resistance increases falls down towards zero, so the fourth derivative with respect to time is probably negative. Once you reach terminal velocity, you can change these derivatives by going spreadeagle, diving headfirst, etc. 17. Suppose you know how many miles per gallon you get while driving. What other information do you need to calculate how much gas you’ll use in the next minute? Answer: your speed 3

18. Suppose P (x) is the percentage of people who live to at least age x. What d is P (0)? What is P (∞)? What is the interpretation of dx P (x)? Answer: d P (0) = 1, P (∞) = 0. dx P (x) is the probability density for age of death. It is highest at the ages people are most likely to die and lowest at the ages people are unlikely to die. 19. When a function is at a local minimum, what can you say about the value, first derivative, and second derivative of the function? Answer: You can’t say anything about its value, except that it has one. The first derivative must be zero, while the second derivative must be positive, or possibly also zero if some higher even derivative is positive.

4