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A practice final exam for the Intuitive Calculus course (Mathematics 11012). The exam covers various topics including derivatives, limits, integrals, and functions. Students are required to find derivatives, identify critical points, draw sign diagrams, calculate limits, and evaluate integrals. The exam also includes problems involving exponentials and logarithms.
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Mathematics 11012 Intuitive Calculus Practice Final Examination R. M. Aron
A. Let f (x) = x 2 x^2 +1.^ Find^ f^
′(x). (The answer need not be simplified.)
B. Consider the curve given by the function g(x) = x^3 − 3 x + 2. (a). Find the tangent line to this curve at x = − 2.
(b). Find all points x at which the tangent to this curve is horizontal.
C. In each case, find the requested derivative: (a). h′(t) where h(t) = tet 2 .
(b). h′(t) where h(t) = (t^4 − 2 t^3 + 6)^5.
1
(c). f ′(x) where f (x) = ln(x^2 + x).
(d). g′′(s) where g(s) = es 2 .
D. Calculate the following expressions: (a). The limit, as n → ∞, of
(4 + (^1) n )^2 − 42 1 n
(b). The limit, as n → ∞, of
n
n
n
n
n
n − 1 n
n
E. Let h(s) = s^3 + 3s^2 − 9 s − 3. (a). Find the critical points of h.
(b). f ′(t), where f (t) = eln(3t (^4) +2t+2) .
(c). g(x) = ex 4
(d). e
(^2) e 3 e−^3 e.
H. Let f (x) = x^2 − 4 x + 6. Compute the extreme (i.e. the biggest and the smallest) values of f on the interval [− 1 , 6].
I. Compute each of the following integrals: (a).
(3x^2 + x − 1)dx.
(b).
∫ (^4) t (^5) +2t (^3) −t (^2) −t t^2 dt.
(c).
− 1 (x
(^3) + 5)dx.
(d).
(x + 2)(2x − 1)dx.
J. Compute each of the following integrals: (a).
(4x^3 − 3 x^2 + 5)^6 (2x^2 − x)dx.
(b).
∫ (^) e 0
x+ x^2 +2x+1 dx.
K. What is the area under the curve y = x^3 and the x−axis, where x varies from 1 to 4.
L.(a). Draw the two curves f (x) = x^2 + 2x − 5 and g(x) = 2x + 4.
O. One bank is offering 20 year certificates of deposit paying 5 % per year, compounded quarterly. Another is offering 20 year certificates of deposit paying 4.5 % per year, compounded continuously. You have $1,000 to invest. In which of the two banks should you deposit your money and why?
P. You buy a brand new Cadillac for $ 50,000. The car depreciates at the rate of 20% per year. When is the car worth half the price you paid?
Q. World consumption of lead is running at the rate of 6.1e^0.^01 t^ million metric tons per year, where t is measured in years, with t = 0 corre- sponding to 2008. Find a formula for the total amount of lead that will be consumed within t years of 2008.
R. A company’s profit from producing x tons of a product is given by P (x) =
x^3 + 2x^2 + 4 thousand dollars (for 0 ≤ x ≤ 10).
(a). Calculate the company’s marginal profit, M P (x).
(b). Calculate P ′(4) and interpret the result.
S. A restaurant manager knows that on a typical day, 100 cheeseburg- ers will be sold at a price of $2.00 each. She also knows that if for each 20 cent reduction in price, the restaurant will sell 25 more cheeseburg- ers. Find the price that the restaurant should charge (and the number of cheeseburgers sold) that will maximize the restaurant’s revenue for cheeseburgers.
