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Avon High School Name _________________________________________ VU Calculus UNIT 5 (3.5-3.7, 3.9) EXAM REVIEW Limits at Infinity/Curve Sketching/Optimization/Differentials Find the vertical asymptote(s), if any. x2 x 6 1.) y x2 4 Find the horizontal asymptote(s), if any. 7x 1 3 x 1 3.) y 2 4.) y 6x 7 5 x 4 2.) y 4 2 x 7 x 15 5.) y 5 x 2 1 3x 2 6.) y 5 x 64 x 2 3 The graph of the function f is shown. Sketch the graph of the derivative, f . 7.) 8.) Analyze and sketch a graph of the given function. x x2 9.) f x 10.) f x 1 x4 x2 1 11.) f x x 9 x 2 Determine the slant asymptote of the graph of f . 12.) f x 2 x 2 6 x 3 x 1 The graph of f is shown to the right. 14.) On what interval(s) is f 0 ? 15.) For which value of x is f zero? 13.) f x x 2 8 x 15 x6 16.) Find two positive numbers whose product is 187 and whose sum is a minimum. 17.) Find the length and width of a rectangle that has an area of 288 square feet and whose perimeter is a minimum. 18.) Find the point on the graph of the function f x x that is closest to the point 18,0 . 19.) Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 121 meters. 5 x (see 2 figure). What length and width should the rectangle have so that its area is a maximum? 20.) A rectangle is bounded by the x - and y -axes and the graph of y 21.) Find the equation of the tangent line T to the graph of f x 23 23 at the given point 2, . 2 x 4 22.) Compare dy and y for y 2 x 2 4 at x 2 with x dx 0.05 . Give your answers to four decimal places. 23.) Compare dy and y for y x 4 4 at x 0 with x dx 0.05 . Give your answers to four decimal places. Find the differential dy of the function. 24.) y x 2 3x 4 25.) y 4 x5 8 26.) y x sin 5x