MATH 1A MOCK FINAL DELUXE PEYAM RYAN TABRIZIAN Name Instructions This is a mock final deluxe designed to give you an idea of what the actual final deluxe will look like Careful The actual final deluxe might have very different questions 1 2 3 4 5 6 7 8 9 10 Bonus 1 Bonus 2 Bonus 3 Total 10 10 10 15 20 15 20 30 10 10 5 5 5 150 Date Friday August 5th 2011 1 2 PEYAM RYAN TABRIZIAN 1 10 points 5 points each Find the following limits a limx b limx x2 x x ln x 2 x MATH 1A MOCK FINAL DELUXE 3 2 10 points Use the definition of the derivative to calculate f 0 x where f x 1 x 4 PEYAM RYAN TABRIZIAN 3 10 points 5 points each Find the derivatives of the following functions a f x xcos x b y 0 where x3 y 3 xy MATH 1A MOCK FINAL DELUXE 5 4 15 points A ladder 10 ft long rests against a vertical wall If the bottom of the ladder is sliding away from the wall at a rate of 1 ft s how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall 6 PEYAM RYAN TABRIZIAN 5 20 points If 12 cm2 of material is available to make a cylinder with an open top find the largest possible volume of the cylinder MATH 1A MOCK FINAL DELUXE 7 6 15 points Show that the following equation has exactly one solution in 1 1 x4 5x 1 0 8 PEYAM RYAN TABRIZIAN 7 20 points Use the definition of the integral to find Z 2 x2 dx 1 You may use the following formulas n X i 1 1 n n X i 1 n n 1 i 2 n X i 1 n n 1 2n 1 i 6 n X i 1 i3 n2 n 1 2 4 MATH 1A MOCK FINAL DELUXE 9 8 30 points 5 points each Find the following a The antiderivative F of f x x2 3x3 4x7 which satisfies F 0 1 b R1 1 x dx Hint Draw a picture 10 PEYAM RYAN TABRIZIAN c R d Re 1 x2 1 1 dx x2 1 ln x 2 dx x MATH 1A MOCK FINAL DELUXE e g 0 x where g x R ex 1 t2 dt x f The average value of f x sin x on 11 12 PEYAM RYAN TABRIZIAN 9 10 points Find the area of the region enclosed by the curves y x2 4 and y 4 x2 MATH 1A MOCK FINAL DELUXE 10 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 points 13 14 PEYAM RYAN TABRIZIAN R1 Bonus 1 5 points Show that if f is continuous on 0 1 then 0 f x dx is bounded that is there are numbers m and M such that Z 1 f x dx M m 0 Hint Use one of the value theorems we haven t used much in this course see section 4 1 MATH 1A MOCK FINAL DELUXE 15 Bonus 2 5 points If f x Ax3 Bx2 Cx D is a polynomial whose coefficients satisfy A B C D 0 4 3 2 Show that f has at least one zero on 0 1 Hint What is the average value of f on 0 1 16 PEYAM RYAN TABRIZIAN Bonus 3 5 points Another way to define ln x is Z x 1 dt ln x 1 t Show 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 ln a
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