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Stanford MATH 19 - Study Notes
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Math 19: Calculus Winter 2008 Instructor: Jennifer KlokePractice Final1. Determine whether each statement is true or false. Unless otherwise stated, any func-tion below is arbitrary. If the statement is true, very briefly cite your reasoning.If it is false, provide an example showing the statement to be false.(a) True or False. If f0(x) < 0 for 1 < x < 6, then f(x) is decreasing on (1,6).(b) True or False. If f(x) has an absolute minimum value at x = c, the f0(c) = 0.(c) True or False. f0(x) has the same domain as f (x).(d) True or False. If f(x) and g(x) are differentiable, thenddx[f(g(x))] = f0(g(x))g0(x)(e) True or False. A function can have two different horizontal asymptotes.(f) True or False. limx→42xx − 4−8x − 4= limx→42xx − 4− limx→48x − 4.12. Complete the following sentence:The function f(x) is continuous on the interval [a, b] if3. Compute the following limits. Justify your results. If the limit is positive or negativeinfinity, it should be clearly indicated instead of just saying the limit does not exist(a) limx→πcos(x + sin(x))(b) limx→12 − x(x − 1)2(c) limx→−414+1x4 + x2(d) limx→−∞2x2− 5x − 2x4− 10x2− 34. Use the limit definition of the derivative to compute the derivative of f(x) =√1 + 2x.35. Compute the following derivatives. You do not have to use the definition of the deriva-tive. If you can “do them in your head” instead of showing every step that is up toyou (though if you get it wrong we cannot give you partial credit.)(a) Let f(x) = 3x ln(x). Find f0(x).(b) Findddxpx ln(x4).(c) Findddx(ln(x))cos(x).4(d) Findddx(4x − 1)3(2x2− 1)3/2(x + 1)2.(e) Findddx(x2cos(x)e3xsin(π/2)).56. Find the equation of the line tangent toy = (2 + x)e−xat the point (0, 2)7. Show that there exists a solution to the equation ln(x) = sin(π2x) on the interval (0, ∞).68. Consider the function f (x) =11 − x2.(a) Write the domain of f(x) in interval notation and find the coordinates of allpoints, if any, where the graph of f crosses the x-axis.(b) Find the equations for all of the vertical and horizontal asymptotes of this func-tion, or state why there are none. Justify asymptotes with limit calculations.7(c) Compute f0(x) and f00(x). You do not have to use the limit definition of thederivative.(d) Determine the intervals on which f(x) is increasing and the intervals on whichf(x) is decreasing.8(e) Determine intervals where f(x) is concave up and the intervals where f(x) isconcave down.(f) Find all the critical points of f(x) and determine which are actually local maximaand which are actually local minima (for each, be sure to justify why it is amax/min or neither.)9(g) Using your answers from parts (a) through (f), Sketch a graph of this function.Be sure to label your axes appropriately.109. For each of the following conditions, provide an example of a function f(x) whichsatisfies the given condition. Unless otherwise indicated, you may express you functioneither with an explicit formula or by a graph.(a) f(x) has neither an absolute minima nor an absolute maxima on its domain.(b) limh→0f(h) = f (0) yet limh→0f(0 + h) − f(0)hdoes not exist.(c) f(x) satisfies f (x) < 0, f0(x) < 0, and f00(x) < 0 for all x.1110. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top).If water is poured into the cup at a rate of 2 cm3/sec, how fast is the water level risingwhen the water is 5 cm deep? (Note: the volume of a cone with height h and radius ris given by V =13πr2h.)1211. Find the equation of the line through the point (3,5) that cuts off the least area fromthe first quadrant. Justify your answer


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Stanford MATH 19 - Study Notes

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