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Math 19: Calculus Summer 2010Practice Final Exam: Winter 20071. (40 points) Finddydxfor each function. Each answer should be a function of x only.(a) (10 points) y =2x − 1−1√x.(b) (10 points) y = (sin x)cos x.(c) (10 points) y =ptan (x2).(d) (10 points) y =(2x + 1)4sin (x2)(ln x)√3x − 1.2. (10 points) Find the equation of the tangent line to the curveex2+ ey2= 2eat the point (−1, 1).3. (20 points) Letf (x) = ln (x2− 1).(a) (10 points) You must show all your work, but please write your final answers in thebox.The domain of f (x) is:f (x) is increasing on:f (x) is decreasing on:f (x) has local maxima at:f (x) has local minima at:f (x) is concave up on:f (x) is concave down on:(b) (4 points) Compute the following four limits.limx→∞ln(x2− 1) = limx→−∞ln(x2− 1) =limx→1+ln(x2− 1) = limx→−1−ln(x2− 1) =1(c) (1 points) List all vertical and horizontal asymptotes of y = ln (x2− 1).(d) (5 points) Using your answers from parts (a) and (b), sketch a graph off (x) = ln (x2− 1).Even if your answers in parts (a) and (b) are wrong, if your sketch correctly usesthose answers, you may earn partial credit.4. (20 points) A particle is moving along the curve x2− 4xy − y2= −11. Given that thex-coordinate of the particle is changing at 3 units/second, how fast is the distance fromthe particle to the origin changing when the particle is at the point (1, 2)? Hint: As anintermediate step, you should compute the value ofdydtwhen x = 1 and y = 2.5. (20 points) A balloon is rising at a constant speed of 1 m/sec. A girl is cycling alonga straight road at a speed of 2 m/sec. When she passes under the balloon it is 3 m aboveher. How fast is the distance between the girl and the balloon increasing 2 seconds later?6. (20 points) A Norman window consists of a rectangle surmountedby a semicircle, as shown. Given that the total area of the windowis A = 8 + 2π, find the minimum possible perimeter of the window.(Please note the horizontal line between the rectangle and the semicir-cle does not count as part of the perimeter.) Hint: The total area hasbeen carefully chosen so that the minimum perimeter occurs at a verysimple value of r. If your optimal value of r is complicated, you havedone something incorrectly.rh7. (20 points) Suppose you have a cone with constant heightH and constant radius R, and you want to put a smaller cone“upside down” inside the larger cone (see figure). If h is theheight of the smaller cone, what should h be to maximizethe volume of the smaller cone? The optimal value of h willdepend on H. Recall that the volume of a cone with baseradius r and height h is given by the formula V =13πr2h.HhRr8. (10 points) For parts (a) and (b), compute the given limits, if they exist. If you assertthat a limit does not exist, you need to justify your answer to get full credit.(a) (5 points) limx→∞(√x2− 3x + 1 −√x2+ 2)(b) (5 points)


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Stanford MATH 19 - Lecture Notes

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