Math 19: Calculus Summer 2010Practice Midterm Problems1. Circle “True” or “False.” No explanation is needed.(a) True False f (x) = |x −2| is one-to-one.(b) True False limx→52xx −5−10x −5= limx→52xx −5− limx→510x −5(c) True False A function can have infinitely many horizontal asymptotes.(d) True False If f is continuous on [0, 2], then f is differentiable on [0, 2].(e) True False The nth derivative of f (x) = e2xis 2ne2x.2. The graph of f (x) is shown. Answer the following questions and explain your reason-ing:(a) What is the domain of f ?(b) What is the range of f ?(c) Is f one-to-one?(d) Where is f not differentiable?(e) Sketch the graph of −f (−x) + 1 on the coordinate system.3. For each of the following limits, evaluate it or show it does not exist.(a) limx→−1x2−3x −4x + 1(b) limx→12ln(sin(π x))(c) limx→2(x2−4)2sin1x − 2(d) limx→∞3 − xx2−3x + 21(e) limx→0f (x), where f (x) =exif x < 0,0 if x = 0,tan2x + 1 if x > 04. Let g (t) =t + 3t −1.(a) Find the equation(s) of all vertical asymptote(s) of g.(b) Find the equation(s) of all horizontal asymptotes of g.(c) Find g−1(t).5. (a) Let f (x) = x2−sin x. Compute f0(x).(b) Show there exists a number a between [0,π2] such that the graph of x2−sin x hasa horizontal tangent line at a.6. (a) Using the limit definition of the derivative, compute the derivative of f (x) =2√x.(b) Find the equation of the tangent line to the curve when x = 1.7. Find the derivatives of the following functions:(a) f (x) = x5− x3/4+ 1(b) f (x) = x ln x(c) f (x) = sin(2ex)(d) f (x) =x2−1x2+ 1(e) f (x) = ln√x cot xex(f) f (x) = |x|8. The displacement (in centimeters) of a particle moving back and forth along a straightline is given by s(t) = 2t+ t3+ 1, where t is measured in seconds.(a) Find the average velocity of the particle from t = 1 to t = 3.(b) Find the instantaneous velocity of the particle at t = 1.(c) Find the acceleration of the particle at t = 1.29. The figure shows the graphs of f , f0, and f00. Identify each curve and explain yourchoices.10. Sketch a possible graph of f (x) which satisfies all of the following conditions:(i) f (0) = 1(ii) limx→−∞f (x) = 0(iii) f0(0) = 1(iv) f is increasing on [−1, 1](v) limx→3−f (x) = 5(vi) limx→3+f (x) = 2(vii) f is decreasing on [3, ∞)(viii) limx→∞f (x) =
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