Calculus Test

Calculus Pretest May 10, 2009

1

Compute 1.

d 3 dx 5x

2.

d dx

sin(x)

3.

d dx

tan

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

0

0

− 4x2 − 3x + 1

1 x




f (t)dt ln(x)f (t)dt

(t(x))

10.

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

16.

d ex −e−x dx 2

arctan(x)

17.

R 2π

18.

R1

0 0

sin(mx) sin(nx)dx

x3 cos(x)dx = −3

1

19.

R∞ 0

e−αx dx

R∞

xe−αx dx (hint: Differentiate both sides of previous problem with re0 spect to α, then set α = 1.) R∞ 21. 0 xn ex dx R∞ R∞ √ 2 2 22. Given −∞ e−x dx = π, find −∞ x2 e−x dx. 20.

23. What is ıı ? 24. Where are the local minima and maxima of x3 e−x ? 25. Find the first three terms in the Taylor series expansion of tan(x) about x0 = π4 .

2

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?”) 2. You’re hiking up a hill. What is the derivative of elevation with respect to position? 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? 4. What is the derivative of the energy needed to compress a gas with respect to the volume it occupies? 5. What is the derivative of distance driven down a road with respect gas in your tank? 6. What is the derivative of money paid at the pump with respect to gas in your tank? 7. What is the derivative of gas in your tank with respect to distance driven? 8. What is the derivative of potential energy with respect to position? 9. What is the derivative of temperature with respect to heat?

10. What is the derivative of an object’s total mass with respect to volume considered? 11. What is the derivative of distance you’ve driven down the road with respect to how many times the wheels have revolved?

2

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? 13. A shark smells blood in the water and is trying to get to it. What does this have to do with derivatives? 14. At the end of a 200m race, is the derivative of a runner’s position (distance from start line) positive or negative? 15. What about the second derivative of position? 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. (The third derivative is called “jerk” and the fourth, fifth, and sixth are called “snap”, “crackle”, and “pop”.) 17. Suppose you know how many miles per gallon you get driving. What other information do you need to calculate how much gas you’ll use in the next minute? 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)? 19. When a function is at a local minimum, what can you say about the value, first derivative, and second derivative of the function?

3

Prove 1. limx→x0 (f (x) + g(x)) = limx→x0 f (x) + limx→x0 g(x) d d (f (x) + g(x)) = dx f (x) + dx g(x) R b d 3. dx F (x) = f (x) ⇔ a dx0 f (x0 ) = F (b) − F (a)

2.

d dx

4.

Pn

5.

Rb

6.

k=1

a

√

k2 =

2k3 +3k2 +k 6

x2 dx = 13 (b3 − a3 )

2 is irrational, that is, there is no pair of integers p, q such that

3

 2 p q

=2