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Calculus i exam ii, fall 2001, which covers topics on derivatives, anti-derivatives, and optimization. Students are required to find derivatives and anti-derivatives of given functions, determine if the volume of a cylinder is increasing or decreasing, and estimate the error of the volume of a square box. The document also includes a problem on maximizing the surface area of a cylinder with a given volume.
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Calculus I, Exam II, Fall 2001 Justify your answers!!
(1) Find the derivative y′^ of the following functions:
(a) f (x) = xπ
(b) f (x) = tan
− (^1) (2x) x^3 +
(c) f (x) = (x^3 + 6x − 4)^25
(d) f (x) = tan(x^5 )
(e) f (x) = ln(sin(x^4 ))
(f) x^4 − y^5 = sec(x^2 y^3 )
1
(2) Find the anti-derivative of the following functions: (a) f (x) = x (^9) − 4 x x^2
(b) f (x) = x sin(x^2 )
(c) f (x) = √ 2 −^42 x 2
(3) If the radius of a round cylinder increases at the rate of 5m/s while the height decreases at the rate of 20m/s, is the volume increasing or decreasing and at what rate, when h = 5m and r = 2m?
(4) If a square box is measured to have sides 7m long with an error of less than. 001 m, use the differential to estimate the error of the volume of the box.
(6) Graph the following function. State all relevant information next to the graph. [You must justify your answers!] y = f (x) = x^2 /^3 (x^2 − 8)
