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241fine

# 241fine - Calculus I(Math 241 Final Problem 1[16 Points...

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Calculus I (Math 241) – Final Problem 1. [16 Points] Calculate the derivatives of the following functions: f ( x ) = sin 3 (2 x - 5) g ( x ) = radicalbig 1 + sec(2 x ) , h ( x ) = sin(2 x ) tan(3 x ) , t ( x ) = radicalbigg 1 + x 1 - x . Problem 2. [16 Points] Calculate the integrals: integraldisplay x ( x - 1) π dx, integraldisplay sec 2 x tan 2 x dx, integraldisplay dt 16 + t 2 , integraldisplay π 3 0 sin 2 x dx. Problem 3. [20 Points] Discuss the function f ( x ) = x 3 - 2 x 2 - 3 x. Specifically, address the following: 1. Find the intercepts and the intervals on which the function is positive, resp., negative. 2. Find the critical points and the intervals on which the function is in- creasing, resp., decreasing. 3. Find the intervals on which the function is concave up, resp., concave down. 4. Find the local maxima and minima and the inflection points. 5. Sketch the graph in accordance with the information obtained above. Problem 4. [14 Points] Use first principles to find f ( a ) if f ( x ) = x 4 . Problem 5. [15 Points] Consider a curve that is defined by the equation y = y (4 - y ) x with y (2) = 3 .

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