Differential and Integral Calculus (Mid_spring2026)

01 a) Explain the limits and continuity of a function (CLO 2, L2) 2 b) Find the indicated limit if it exists: Lim_{x->-2} (x^2-x-6)/(x^2+3x+2) (CLO 3, L3) 2 c) Graph the function: f(x)=1+x^2 (CLO 3, L3) 3 02 Find dy/dx a) y = 10x^4 + 10/x^2 - 12x + 502 (CLO 3, L3) 2 b) y = 100(3x^2 + 5)^4 (CLO 3, L3) 2 c) y = 10x / (1 + x) (CLO 3, L3) 2 03 a) Records indicate that x years after 2005, the average property tax on a three-bedroom home in a certain community was T(x) = 20x^2 + 40x + 600 dollars. i. At what rate was the property tax increasing with respect to time in 2005? ii. By how much did the tax change between the years 2005 and 2009? (CLO 3, L3) 3 b) When a certain commodity is sold for p dollars per unit, consumers will buy D(p)=40,000/P units per month. It is estimated that t months from now, the price of the commodity will be p(t)=0.4t^(3/2) + 6.8 dollars per unit. At what rate will the monthly demand for the commodity be changing with respect to time 4 months from now? (CLO 3, L3) 3 04 a) Distinguish between marginal profit and actual profit with an example (CLO 4, L4) 3 b) A manufacturer estimates that when x units of a particular commodity are produced, the total cost will be C(x) = (1/8)x^2 + 3x + 98 dollars, and furthermore, that all x units will be sold when the price is p(x) = (1/3)(75-x) dollars per unit. i. Find the marginal cost and the marginal revenue. ii. Use marginal cost to estimate the cost of producing the ninth unit. iii. What is the actual cost of producing the ninth unit? iv. Use marginal revenue to estimate the revenue derived from the sale of the ninth unit. v. What is the actual revenue derived from the sale of the ninth unit? (CLO 3, L3) 3