Calculus Answers

A 10 ft plank is leaning against a wall. If at a certain instant the bottom of the plank is 2 ft from the wall and is being pushed toward the wall at the rate of 6 in/s , how fast is the acute angle that the plank makes with the ground increasing?

Show using an example 𝑓 and 𝑔 are not integrable on [𝑎, 𝑏], but 𝑓𝑔 may be integrable on [𝑎, 𝑏].

1. A plant grows 1.67 cm in its first week. Each week it grows by 2% more than it did the week before. By how much does it grow in twenty weeks, including the first week?

2.  Two functions f and g are defined on the set R, of real numbers by

f:x-> 2x - 3 or f:x-> 2x-3/x

x

and

g : x -> 1- 3x OR g:x-> 1- 3x/2 Find the inverse of the function f and the domain of gof.

2

1. From question b above, As a farmer, can you come out with a general term that can be used to forecast the height of the plant at any particular time? Explain mathematically.
2. A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.

Use double integration to find the area of the plane region enclosed by the given curves. y² = 324 - x and y² = 324 – 324x.

 A demographer estimates that the population of inmates at the Nsawam prisons will increase at the rate of 298𝑒0.8𝑥 per year in x years. Currently there are about 7800 inmates at the prison. How many inmates should the prison authorities expect 10 years from now?

Consider a rectangle with perimeter 28 (units). Let the width of the rectangle be w (units) and let the Area of the region enclosed by the rectangle be A ( square units). Express A as a function of w and state the domain and range of the function.

e lengths p, q, and r of the edges of a rectangular box are changing with time. At the

instant p = 2m, q = 3m, r = 4m,

dp

dt =

dq

dt = 1 m/sec and dr

dt = −2 m/sec. At what rate is the

box’s volume V changing at that instant? 

A cellular phone company has the following production function for a smart phone: p(x, y) =

50x

2

3 y

1

3 where p is the number of units produced with x units of labor and y units of capital. a)

Find the number of units produced with 125 units of labor and 64 units of capital. b) Find the

marginal productivities (Hints: Partial derivatives). c) Evaluate the marginal productivities at

x = 125 and y = 64.