MATH 1A - MOCK FINAL DELUXEPEYAM RYAN TABRIZIANName:Instructions: This is a mock final deluxe, designed to give you an ideaof what the actual final deluxe will look like.Careful! The actual final deluxe might have very different questions!1 102 103 104 155 206 157 208 309 1010 10Bonus 1 5Bonus 2 5Bonus 3 5Total 150Date: Friday, August 5th, 2011.12 PEYAM RYAN TABRIZIAN1. (10 points, 5 points each) Find the following limits(a) limx→∞√x2+ x − x(b) limx→∞(ln(x))2xMATH 1A - MOCK FINAL DELUXE 32. (10 points) Use the definition of the derivative to calculate f0(x),where:f(x) =1x4 PEYAM RYAN TABRIZIAN3. (10 points, 5 points each) Find the derivatives of the following func-tions(a) f(x) = xcos(x)(b) y0, where x3+ y3= xyMATH 1A - MOCK FINAL DELUXE 54. (15 points) A ladder 10 ft long rests against a vertical wall. If thebottom of the ladder is sliding away from the wall at a rate of 1ft/s, how fast is the top of the ladder sliding down the wall when thebottom of the ladder is 6 ft from the wall?6 PEYAM RYAN TABRIZIAN5. (20 points) If 12π cm2of material is available to make a cylinderwith an open top, find the largest possible volume of the cylinder.MATH 1A - MOCK FINAL DELUXE 76. (15 points) Show that the following equation has exactly one solu-tion in [-1,1]x4− 5x + 1 = 08 PEYAM RYAN TABRIZIAN7. (20 points) Use the definition of the integral to find:Z21x2dxYou may use the following formulas:nXi=11 = nnXi=1i =n(n + 1)2nXi=1i =n(n + 1)(2n + 1)6nXi=1i3=n2(n + 1)24MATH 1A - MOCK FINAL DELUXE 98. (30 points, 5 points each) Find the following:(a) The antiderivative F of f(x) = x2+ 3x3−4x7which satisfiesF (0) = 1(b)R1−1|x|dx (Hint: Draw a picture)10 PEYAM RYAN TABRIZIAN(c)Rx2+ 1 +1x2+1dx(d)Re1(ln(x))2xdxMATH 1A - MOCK FINAL DELUXE 11(e) g0(x), where g(x) =Rexx√1 + t2dt(f) The average value of f(x) = sin(x) on [−π, π]12 PEYAM RYAN TABRIZIAN9. (10 points) Find the area of the region enclosed by the curves:y = x2− 4 and y = 4 − x2MATH 1A - MOCK FINAL DELUXE 1310. (10 points) If f(x) = x ln(x), find:(a) Intervals of increase and decrease, and local max/min(b) Intervals of concavity and inflection points14 PEYAM RYAN TABRIZIANBonus 1 (5 points) Show that if f is continuous on [0, 1], thenR10f(x)dx isbounded, that is, there are numbers m and M such that:m ≤Z10f(x)dx ≤ MHint: Use one of the ‘value’ theorems we haven’t used much inthis course (see section 4.1)MATH 1A - MOCK FINAL DELUXE 15Bonus 2 (5 points) If f(x) = Ax3+ Bx2+ Cx + D is a polynomial whosecoefficients satisfy:A4+B3+C2+ D = 0Show that f has at least one zero on [0, 1].Hint: What is the average value of f on [0, 1]?16 PEYAM RYAN TABRIZIANBonus 3 (5 points) Another way to define ln(x) is:ln(x) =Zx11tdtShow using this definition only that ln(ab) = ln(a) + ln(b).Hint: Fix a constant a, and consider the function:f(x) = ln(ax) − ln(x) −
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