Review Problems on Derivatives Brief solutions are posted in the web syllabus together with the daily assignment for the first day of class 1 Find the derivative C B 0 w B a C 0 B sin cos tan B b C 0 B ln sec B c C 0 B arctan d C 0 B log B B B e C 0 B Barcsin B f C 0 B arctan B B g C 0 B ln lsin Bl h B C sin C B i BC CB 2 The Mean Value Theorem see text pp 280 282 for a review states If 0 is a differentiable function on the interval then there exists a number between and such that 0 w 0 0 or equivalently 0 w 0 0 The Mean Value Theorem is very important not so much for computations as a tool in understanding other useful tools and facts in calculus a Draw a picture of a continuous function with domain some interval Draw the straight line segment P joining the points 0 and 0 in your picture What is its slope in terms of and 0 b 0 w represents the slope the tangent line to the graph at the point 0 Find a point in your picture where the equation is true What does the equation say geometrically c Suppose we have a car moving along a straight highway Its position in km at time B hours is given by the function C 0 B What does the right hand side of equation represent physically what does the left hand side represent physically d Suppose you drive from here to Kansas City along a straight line highway more or less like I 70 Your average velocity is 100 km hr What does the Mean Value Theorem tell you happened at some time during the trip 3 Suppose we have a function C 0 B for which 0 w B B B a Where is the graph of 0 B increasing decreasing b Where does 0 B have a local maximum or minimum c Where is the graph of C 0 w B increasing decreasing d Where is the graph of 0 B concave up concave down e Where does 0 B have inflection points 4 The table bleow gives the position 0 in feet of a point moving along a line after seconds a Based on this data what is the best estimate you can make of the velocity of the point at time b The answer to a is your estimate to the value of what derivative
View Full Document
Unlocking...