Review Problems on Derivatives(Brief solutions are posted in the web syllabus together with the daily assignment for thefirst day of class.)1. Find the derivative ( =.C.Bw0ÐBÑÑ a) sin cos tanCœ0ÐBÑœ Ð Ð BÑÑ b) ln secC œ 0ÐBÑ œ Ð ÐB "ÑÑ## c) Cœ0ÐBÑœ"!arctanÐ# ÑB d) logC œ 0ÐBÑ œ ÐB/ Ñ(B"È$# e) Cœ0ÐBÑœBarcsin ÐB Ñ# f) Cœ0ÐBÑœarctan B"B# g) ln sinC œ 0ÐBÑ œ l Bl h) sinBC CœB## i) BœCCB2. The Mean Value Theorem (see text, pp. 280-282 for a review) states: If is a differentiable function on the interval , then there exists a0Ò+ß,Ó number between and such that-+, (*) 0Ð-Ñœw0Ð,Ñ0Ð+Ñ,+ (or, equivalently, 0 Ð-ÑÐ, +Ñ œ 0Ð,Ñ 0Ð+ÑwThe Mean Value Theorem is very important not so much for computations as a tool inunderstanding other useful tools and facts in calculus. a) Draw a picture of a continuous function with domain some interval Ò+ß ,ÓÞDraw the straight line segment joining the points and in yourP Ð+ß 0Ð+ÑÑ Ð,ß 0Ð,ÑÑpicture. What is its slope (in terms of and ) ?+ß ,ß 0 b) represents the slope the tangent line to the graph at the point .0 Ð-Ñ Ð-ß 0Ð-ÑÑwFind a point in your picture where the equation (*) is true. What does the equation saygeometrically?c) Suppose we have a car moving along a straight highway. Its position (in km) attime hours is given by the function . What does the right hand side ofBCœ0ÐBÑequation (*) represent physically? what does the left hand side represent physically? d) Suppose you drive from here to Kansas City along a straight linehighway more or less like I-70. Your average velocity is 100 km/hr. What does theMean Value Theorem tell you happened at some time during the trip?-3. Suppose we have a function for whichCœ0ÐBÑ 0 ÐBÑ œ ÐB "ÑÐB #Ñw# a) Where is the graph of increasing? decreasing?0ÐBÑ b) Where does have a local maximum or minimum?0ÐBÑ c) Where is the graph of increasing? decreasing?C0 ÐBÑw d) Where is the graph of concave up? concave down?0ÐBÑ e) Where does have inflection points?0ÐBÑ4. The table bleow gives the position (in feet) of a point moving along a line=œ0Ð>Ñafter seconds.> >!"#$=!$%'a) Based on this data, what is the best estimate you can make of the velocity of the pointat time ?>œ#b) The answer to a) is your estimate to the value of what
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