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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
6. A company manufacturing wicker baskets knows that it has variable costs of $0.62 per unit and fixed costs of $150 per week. Its baskets sell for $2.40 each. Assuming that total costs are a linear function of output, calculate the level of weekly sales necessary to:
(i) Break-even
(ii) Reach a profit of $4300 per week
5. Suppose you plan to set up a firm to retail a certain product A. From a market survey, you estimate that the annual sales volume is 20,000 units at a selling price of $25 per unit. You can purchase product A from a manufacturer at a cost of $13 per unit. Total fixed cost, which includes rental, depreciation, etc. is estimated at $180,000 per annum.
(a) Calculate the break-even point in units, in dollars.
(b) Calculate the profit based on the expected sales volume.
