Calculus Answers

Given a sequence a_n=((n!)²)/(2n)! . Evaluate lim n approaches infinity a_n

Find fx(x,y),fy(x,y),fx(1,3)and fy(-2,4) for the given function. If z=f(x,y)=3x3y2-x2y3+4x+9

Find two level curve of f(x,y)=2xy/(x^2+y^2 ) . Give a rough sketch

Find volume bounded by the planes 𝑥 = 0, 𝑦 = 0, 𝑧 = 0 and 2𝑥 + 𝑦 + 𝑧 = 6 0 ≤ 𝑥 ≤ 2,0 ≤ 𝑦 ≤ 6 − 2�

3) The mean value theorem for differentiation states that if f(x) is continuous on a closed

interval [a,b] and differentiable on the open interval (a,b), then there exist a number c

in (a,b) such that

f'c=f(b)-f(a)/b-a

If f (x) =5x^2+ 3x 2 , find the value of c in the interval (2,4) that satisfies the above

theorem.

4)The mean value theorem for Differentiation states that if f(x) is differentiable on (a, b)and

continuous on [a, b], then there is at least one point c in (a, b) where f'(c)=f(b)-f(a)/b-a

Find the value of c that satisfies the theorem for the function f(x) = x 6 + in the interval [-2,

10].

1) A designer of a box making company wants to produce an open box from a piece of card

with the width of 16cm and the length of 20cm. The box is made by cutting out squares of

length x at each corners.

i)Show that the volume of the box is V=4x^3- 72x^2-320x

ii) Find the value of x so that the volume of the box is maximum.

iii) Find the dimensions of the box so that the volume of the box is maximum.

2) A glass window frame is in the form of a rectangle at the bottom and a semicircle at the

top as shown in Figure 1, has a perimeter of 4 m.

i) Show that the area of the glass window is given by

A = 2x − (4 + π8 )x^2

ii) Find the values of x and y that will maximized the area of the glass window.

(0.56 m)

iii) Hence, find the maximum area of the glass window.

Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If 

𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥ଷ𝑦ଶ − 𝑥ଶ𝑦ଷ + 4𝑥 + 9

15

A firm’s production function is given by

Q = 700Le^−0,02L,

where Q denotes the number of units produced and L the number of labourers. Determine the size of the  workforce that maximises output.

[a.   L = 14

[b.   L = 50

[c.   L = 328

[d.   L = 700

13

Given the function f (x) = x⁴ − 2x² + 2,

the second derivative of f is

[a.      −10

[b.   10

[c.   12x² − 4

[d.   2x(x − 5)

Question 14

Customers of a hardware store are willing to buy

Q = 80 − P ²

boxes of nails at P rand per box. Find the marginal revenue if the price per box is R3.

a.       R−53.00

b.       R71.00

c.       R53.00

d.       R−71.00

11

The marginal labour cost function is given by the equation

MLC = 3 + 4L.

Calculate the cost of employing the first eight labourers.

a.       −114

b.       114

c.       152

d.       119

12

If the demand function is

P = Q² − 7Q − 8,

where P and Q are the price and quantity respectively, determine the point price elasticity of demand (rounded to three decimal places) if P = 51. Is demand elastic or inelastic at this price?

[a.   e_d = −0.075;inelastic [b.  e_d = −0.255;inelastic

[c.   e_d = 1.300; elastic

[d.  e_d = −0.253; inelastic