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Unformatted text preview: Math 180 Final Exam Show all your work. An unjustified answer is not correct. May 5, 2005 1. Find the derivative of the following functions, do not simplify. (a) x2005 + x2/3 , (b) cos(x), (c) 1 + 2x . 3 + x2 2. Use calculus to find the exact x and ycoordinates of any local maxima, local minima, and inflection points of the function f (x) = x3  3x + 2. 3. Use implicit differentiation to find the slope of the line tangent to the curve x2 + xy + 2y 2 = 4 at the point (1, 1).
100 4. Estimate the integral
0 f (t) dt using the left Riemann sum with five subdivisions. Some values of the function f are given in the table: t f (t) 0 1.2 20 2.8 40 4.0 60 4.7 80 5.1 100 5.2 If the function f is known to be increasing, could the integral be smaller than your estimate? Why or why not? 5. Write an integral which gives the area of the region between x = 1 and x = 3, above 1 the xaxis, and below the curve y = x  2 . x Evaluate your integral to find the area. 6. Find the average value of the function f (x) = 4  x2 on the interval 0 x 3. 7. Find
x0 lim 1+x1 . x Explain how you obtain your answer. 8. Differentiate (a) x2 e3x , (b) arctan(x), 1 (c) ln(cos(x)). 9. The function f (x) has the following properties: f is increasing, f is concave down, f (3) = 2, f (3) = 0.5. (a) Find the tangent line to y = f (x) at the point (3, 2). (b) Use (a) to estimate f (3.3). (c) Sketch a possible graph of f (x) for 1 x 4. (d) Could your estimate in (b) be greater than f (3.3)? 10. A family of rectangles in the xyplane have one side on the xaxis, the lower left corner at the origin (0, 0), and the upper right corner at a point (x, y) on the straight line 3x + 4y = 5. (a) Find the area of such a rectangle as a function of x alone. (b) Find the dimensions, x and y, of the particular rectangle with the largest area. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ............................. ............................ .. .. . .... . ..... . ... ... . . ... ... . . ... ... . . ... . ... . . . . (x, y) 2 ...
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 Spring '08
 TAN
 Calculus, Derivative

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