Practice Problems

Question 1

Limit of (cosx)^(1/(x^2)) as x approches zero from the positive direction?

Answer 1

1/(e^0.5)

Question 2

Limit of xsin(pi/x) as x approaches infinity?

Answer 2

pi

Question 3

Limit of x^(1/(1-x)) as x approaches one from the positive direction?

Answer 3

e

Question 4

A man is in a river three kilometers wide, and needs to get to a point 8 kilometers to his left and on the opposite bank. He can paddle at 6 km/hr, and jog along the shore at 8 km/hr. What is the optimized distance for him to run along the shore?

Answer 4

8 - 9/(7^0.5) meters

Question 5

Limit of cotx - (1/x) as x approaches zero from the positive direction?

Answer 5

0

Question 6

Limit of ((2x-3)/(2x+5))^(2x+1) as x approaches infinity?

Answer 6

1/(e^2)

Question 7

Find two positive numbers whose product is 100 and whose sum is a minimum.

Answer 7

10, 10

Question 8

A model is used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units is: Y = (kN)/(1+n^2), in which k is a positive constant. What nitrogen level gives the best yield?

Answer 8

N = 1

Question 9

A farmer with 750 ft of fencing wants to make four pens, and optimize the maximum area within the four collectively. (a) Write an equation for the situation, (b) write an expression for the total area, (c) write a function for the total area using only one variable, and (d) solve for the optimized length for both variables.

Answer 9

(a) 5x + 2y = 750
(b) xy = A
(c) A(x) = 375x - (5/2)x^2
(d) x = 75 ft; y = 187.5 ft