Calculus Answers

a) Define differentiation and integration in calculus. Also write down the 

 differences between them.

3. a) Define tangent and normal of a curve with figure. Also find the equation of tangent and normal of the ellipse (x ^ 2)/4 + (y ^ 2)/16 = 1 at the point (- 1, 3) .

b) Explain maximum and minimum value of a function with graphically. Evaluate maximum and minimum value of the function f(x) = x ^ 3 - 3x ^ 2 + 3x + 1

5. a) Calculate the area under the curve y = x ^ 3 + 4x + 1 from x = - 3 to x = 3 .

b) Evaluate: integrate x * e ^ x dx from 0 to 4

4. a) Evaluate : integrate (x ^ 3 + 1/(x ^ 3)) dx

b) int 0 ^ pi 2 -4 sin xdx 3+4 cos x

2. a) Find the derivatives of the following functions with respect to x.

x ^ 3 + y ^ 3 = 3

y = (sin x) ^ tan x

b) Evaluate the 2 ^ (nd) order partial derivatives partial^ 2 u partial x^ 2 and partial^ 2 u partial y^ 2 if u=2x^ 3 +3x^ 2 y+xy^ +y^ .

5. a) Calculate the area under the curve y = x ^ 3 + 4x + 1 from x = - 3 to x = 3

b) Evaluate: integrate x * e ^ x dx from 0 to 4

1. a) Define differentiation and integration in calculus. Also write down the differences between them.

b) Write down some application of Calculus in CSE.

c) Describe geometrical meaning of definite integral with figure.

1.   You wanted to start your business of manufacturing custom made mobile covers for the newest model of iPhone. The selling price is  per cover. The cost function to manufacture the covers is  where  is the number of covers sold. Show at least 5 decimal places

a.   Find the profit function . 3 Marks

b.     Find the quantity  that maximizes profit. 2 Marks

c.     Explain the steps you take to arrive at your answer in part b. 3 Marks

d.     Interpret the answer. 2 Marks

Differentiate the following functions.(use Chain Rules)

1. y=(5x² —2x + 1)—³

A small factory producing a single product has weekly fixed costs of production of $2,112 and weekly variable costs of $52x + 3/4 x2, where x is the quantity produced. the capacity of the factory is about 600 units.

Past experience suggests that the product’s price and quantity are linked by the following demand equation: p = 200 - 1/4 x (p, x > 0) where p = $ price/unit and x = quantity sold. You are required to:

(a) Find the level of production at which revenue is maximized

(b) Find any break-even points