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A series of calculus problems covering topics such as limits, derivatives, the mean value theorem, implicit differentiation, and optimization. It includes problems on evaluating limits using graphs, computing derivatives of various functions, applying the mean value theorem, finding derivatives using implicit differentiation, and solving optimization problems related to rates of change and finding maximum/minimum values. The document also includes a problem on linear approximation and sketching a graph based on given conditions. This material is suitable for students studying calculus at the university level, providing a comprehensive review of key concepts and problem-solving techniques. It is designed to enhance understanding and proficiency in calculus through practical application and critical thinking.
Typology: Exams
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Problem 1: (5 points each) a.^ Use^ the^ graphs^ to^ evaluate the following^ limits:
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lim^ f^ )^ /NE
b.^ Compute^ the^ limit,^ show and justify^ your^ steps. "^ lim2^4 _+|';I L^ r—+- T y ¥^ / T/l^ NS^ z i I^2 Y-
Problem^ 2:^ (4^ points^ each)^ For^ each^ function^ f(x),^ compute^ the derivative^ f'(x).^ No^ need^ to^ simplify^ your^ answers. f(x) = sinh(In(x)) Cosh^ (W^ ()^ () FOO^ =^ 4(x7^ -^ 314!^14 ^ W^ ATE ()‘7’5"”)/);}'2&“) X^ ~3X'-3x^ ¢ X'q‘b‘(“fi’q‘g |^ ] Y^ W_^ oy 4 ue® 3()(7'3“)( -12%) e x f(x)—2x2+x+ (425^ + s^ D^ () (Y (th)fl^ !)z £(x)^ =^ cos(x?)^ e?*
.^24 L)^ 20+ (ol
arefen(20)^ o^ m‘;)
changed when the side lengths are 2 cm? Provide units in your answer. 75 Z Z v /S¢¢ Problem^ 6:^ (10^ points)^ For^ the^ function^ f(x)^ =^ xIn(x)^ .. a. Find the critical numbers. b. Determine^ the^ intervals^ on^ which^ f(x)^ is^ increasing^ and^ decreasing. -\
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Problem 7:^ (10^ points)^ Answer^ ONLY^ ONE^ of^ the^ following^ questions: =^ f: )^ o^ x)=1+cos(x Use^ the^ closed^ interval^ method^ to find the^ maximum^ and^ minimum^ values of f^ ()^ (x)^ on^ the interval [— %nl.^ sy^ f a box that re requi b. Abox has^ a^ square^ base,^ an^ open^ top,^ and^ a^ volume^ of 32 m?.^ Find^ the^ dimensions^072 aie the least^ amount^ of^ material. Ll^ s^ -sin(0^ Ate)e^ 14eostr)^ ‘fll/^ i 7 e W (^) ' e yz(')'f'fl' Druy,^ W^ et^ (iniwiahs & N iviweam = (P) Problem 8: (10 points) Use either^ linear^ approximation^ OR^ differentials^ to^ estimate^ the^ quantity^ V101. i\ NN (x-0)+o 4 (100) ‘e
