Math 19: Calculus Summer 2010Midterm Exam — July 20, 2010, 7:00 to 9:00 PMName:Section (circle one): Eric · 1:15 PM Anca · 2:15 PM• You have a maximum of 2 hours. This is a closed-book, closed-notes exam. Nocalculators or other electronic aids are allowed.• Read each question carefully. Show your work and justify your answers for fullcredit. You do not need to simplify your answers unless instructed to do so. Youmay use results from class, but if you cite a theorem you should check that thehypotheses are explicitly verified.• If you need extra room, use the back sides of each page. If you must use extra paper,make sure to write your name on it and attach it to this exam. Do not unstaple ordetach pages from this exam.• Please sign to indicate that you have read and agree to the following statement:“On my honor, I have neither given nor received any aid on this examina-tion. I have furthermore abided by all other aspects of the Stanford HonorCode with respect to this examination.”Signature:Grading1 /8 6 /102 /12 7 /103 /10 8 /104 /10 9 /105 /10 10 /10Total /10021. (8 points) Circle “True” or “False.” No explanation is needed.(a) True False If f (x) is not defined at x = 1, then limx→1f (x) does not exist.(b) True False The nth derivative of exis ex.(c) True False limx→2x2−4x − 2does not exist because00is an indeterminate form.(d) True False The sum of two continuous functions is continuous.(e) True False A function can have two different horizontal asymptotes.(f) True False If p is a polynomial function, then limx→ap(x) = p(a).(g) True False If f and g are differentiable, thenddx[ f (x)g (x)] = f0(x)g0(x).(h) True Falseddx(cos x) = sin x.32. (12 points) Evaluate the following limits, or show they do not exist:(a) (3 points) limt→01t√1 + t−1t(b) (3 points) limx→2x2−5x + 6x2+ 2x −84(c) (3 points) limw→2πln(cos w)(d) (3 points) limu→πcos u(u − π)253. (10 points)(a) Complete the following definition: f (x) is continuous at x = a if(b) Find the value of c so that the functionf (x) =(cx2+ 2x if x < 2,x3− cx if x ≥ 2is continuous for all real x.64. (10 points) Show that the equation x + ln x =2xhas a solution c in the interval (1, 2).75. (10 points) Let f (x) = e1/x.(a) (3 points) Find the equation(s) of all vertical asymptote(s), or explain why none exists.Justify your answer with limit calculations.(b) (3 points) Find the equation(s) of all horizontal asymptote(s), or explain why noneexists. Justify your answer with limit calculations.8(c) (2 points) f (x) is a one-to-one function. Find an expression for its inverse function,f−1(x), in terms of x.(d) (2 points) Find the domain of f−1(x).96. (10 points)(a) (3 points) State a definition of the derivative of f (x) at x = a:f0(a) =(b) (4 points) Use your definition to compute the derivative of the function f (x) =1x − 2at x = 3.(c) (3 points) Find the tangent line to the graph y = f (x) at the point (3, 1).107. (10 points) Compute the derivatives of the following functions. Show your work forfull credit.(a) (3 points) f (x) = x5+ x−3ex(b) (3 points) r =sec θ1 + sec θ(Hint: sec θ =1cos θ.)(c) (4 points) y = (1 + 2w)10118. (10 points) A particle moves along a horizontal line, with its position given by thefunction s(t) = t3−3t2+ 2t, where s is in centimeters and t is in seconds.(a) (2 points) Find the average velocity of the particle from t = 1 to t = 2.(b) (4 points) Find the instantaneous velocity v(t) of the particle.(c) (4 points) Find the acceleration a(t) of the particle.129. (10 points) For what values of a and b is the line 2x + y = b tangent to the parabolay = ax2when x = 2?1310. (10 points) On the coordinate axes below, sketch the graph of a function f (x) with thefollowing properties:(i) limx→−∞= ∞(ii) limx→∞= 2(iii) f (x) has a jump discontinuity at x = −2(iv) f (x) is continuous but not differentiable at x = 4(v) f (x) has a vertical asymptote at x = 0(vi) f (x) is continuous from the right at x = 0(vii) f0(1) = −1(viii) f (2) = 1(ix) f0(2) = 0(x) f0(3) = 2xy = f (x)−3 −2−1 12