Differential Calculus 2015

1. For the following problems find f’(x): (i) 𝑓(𝑥) = 𝑥 3 − 1 10 𝑥

(ii)

𝑓(𝑥) =

(iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

𝑓(𝑥) = 𝑎𝑥 + 𝑏 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 𝑓(𝑥) = (10 − 𝑥 5 )(𝑥 6 − 2𝑥 3 + 𝑥) 𝑓(𝑥) = (5 − 𝑥 + 3𝑥 2 )(𝑥 8 + 5𝑥 3 − 10𝑥 2 ) 𝑓(𝑥) = 𝑥 3 ln⁡𝑥 𝑓(𝑥) = [ln⁡(𝑥 2 + 1)]4 𝑓(𝑥) = [(𝑥 − 5)√𝑥 + 4]/(1 − 𝑥) 𝑓(𝑥) = [(𝑥 2 + 3)/(4 − 𝑥)]3

(xi)

𝑦 = 𝑓(𝑢) = 2−𝑢2 𝑢 = 𝑔(𝑥) = 𝑥 2 + 2

(xii)

𝑦 = 𝑓(𝑢) = √𝑢2 − 5𝑢 and 𝑢 = 𝑔(𝑥) = 4/𝑥

6𝑢−1

2. If 𝑧 = 𝑥 2 + 7𝑥𝑦 + 2𝑦 2 + 2𝑥 + 5𝑦 + 7 then find

𝑑𝑧

, 𝑑𝑥

𝑑𝑧

𝜕2 𝑧

,⁡ , 𝑑𝑦 𝜕𝑥 2

𝜕2 𝑧 𝜕𝑦 2

𝜕2 𝑧

𝜕2 𝑧

, 𝜕𝑥𝜕𝑦 and , 𝜕𝑦𝜕𝑥

3. The population of a city is estimated by the function 𝑃 = 𝑓(𝑡) = 1.2𝑒 0.045𝑡 . where 𝑃 equals the population (in millions) and 𝑡 equals time measured in years since 1988 (i) What is the population expected equal to 1995? (ii) Determine the expression for the instantaneous rate of change in the population? (iii) At what rate of population expected to be changing in 1995? 4. Following a rapid increase in the value of residential homes during the mid 1980s,real estate values in Northeast began to drop in 1990. The function 𝑉 = 𝑓(𝑡) = 140000𝑒 −0.002𝑡 is a function which estimates the average value 𝑉 (in$) of a single family residence in one particular township, where 𝑡 equals time measured in months since January 1, 1990. (I) What was the average value estimated to equal on July 1, 1990? On January 1, 1992? (II) Determine the general expression for the instantaneous rate of change in the average value of single residence in this town? (III) At what rate is the value changing on January 1, 1991? 5. A local travel agent is organizing a charter flight to a well-known resort. The agent has quoted a price $ 300 per person if 100 or fewer sign up for the flight. For every person over the 100, the price for all decrease by $ 2.50. For instance, if 101 people sign up each will pay $ 297.50. Let 𝑥 equal the number of persons above 100. (I) Determine the function which states price per person 𝑝 as a function of 𝑥, or 𝑝 = 𝑓(𝑥). (ii) Formulate the function 𝑅 = ℎ(𝑥), which states total ticket revenue 𝑅 as a function of 𝑥.

(iii) What is the value of 𝑥 results in maximum value of 𝑅? (iv) What is the maximum value of 𝑅? (v) What price per ticket results in the maximum value of 𝑅? 6. The total cost of producing 𝑞 units in a certain product is described by the function 𝐶 = 12500000 + 100𝑞 + 0.02𝑞2 (i) (ii) (iii)

Determine how many units 𝑞 should be produced in order to minimize the average cost per unit What is the minimum average cost per unit? What is the total cost of production at this level of output?

7. The quadratic total cost function for a product is 𝐶 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑥 equals the number of units produced and sold and 𝐶 is stated in dollars. The product sells at a price 𝑝 dollars per unit. (i) Construct the profit function stated in terms of 𝑥. (ii) What value of 𝑥 results in maximum profit? (Iii) What restriction assures that a relative maximum occurs at this value of 𝑥? (iv)

What restrictions on 𝑎, 𝑏, 𝑐 and 𝑝 assure that 𝑥 > 0?

8. The utility function of a consumer is 𝑈 = 𝑥𝑦 and the total income is Rs 100. The price of commodity A is Rs 2 per unit and commodity B is Rs 5 per unit. 𝑥 and 𝑦 are the units purchased of commodity A and B respectively. Find the values of 𝑥 and 𝑦 3𝑝

9. The price elasticity of a commodity is 𝑑 = (𝑝−1)(𝑝+2) . Find the corresponding demand function when quantity demanded is 8 units and the price is Rs 2. 10. The demand and supply functions under perfect competition are 𝑦 = 16 − 𝑥 2 and 𝑦 = 2𝑥 2 + 4. Find the market price, producer’s surplus and consumer’s surplus.