Final Exam Review Math Determine the derivative for each of the following: dy dx. dy dx. dy dx dy dx. dy dx dy dx. dy dx

Transcription

1 Final Eam Review Math. Determine the derivative or each o the ollowing: a. y 6 b. y sec c. y ln d. y e. y e. y sin sin g. y cos h. i. y e y log j. k. l. 6 y y cosh y sin m. y ln n. y tan o. y arctan e p. q. y cos y r. y cos s. y tan tan t. y ln tan. Find or y sin.

2 . Find an equation or the tangent line at. Find d y or y e.. or the unction. Find or y / 6. Find or y y. 7. At a distance o 6 t rom the launch site, a spectator is observing a space shuttle being launched. I the space shuttle lits o vertically, at what rate is the distance between the spectator and shuttle changing at the instant when the angle o elevation is and the shuttle is traveling at 6 88 t/sec? 8. I y, use the deinition o derivative to ind. 9. Find the rectangle o largest area that can be inscribed in a semicircle o radius R, assuming one side o the rectangle lies on the diameter o the semicircle as shown. R. I, ind. What is the domain o. Are the lines y and y perpendicular? then a?. True or False: I ( ) L L. a d. I and g are dierentiable, then g.. Sketch a unction where is continuous at a but is not dierentiable at a.. I h( ) [ g( )], g(-), g'(-), (), and '() -, ind an equation o the tangent line to the graph o h at. 6. Find the maimum and minimum values o A B [a, b]. where A and B are constants on 7. Find sin. 8. Does the unction g satisy the hypotheses o the Mean Value Theorem on [,].

3 9. By the theorem, i is continuous on the interval [a, b] and K is between and K c or some c in (a, b). a b then ln up and concave down. Find all inlection points.. Let. Calculate " and use it to ind intervals where the graph o is concave. Find the equation o the tangent line to the graph o the equation tan( y) y at (/, ).. Consider the unction whose graph is a. What is the value o b. ' ( ) at. c. " or < < d. ' ails to eist at. e. ails to be continuous at.. ( ) ails to eist at =. o. Evaluate the ollowing its a. 7 b. e

4 c. sin ln d. sin e.. sin tan /. For the graph o y, ind the inlection point.. Does the graph o sin y have a vertical asymptote at? 6. Veriy that the unction ( ) satisies the hypotheses o the Mean Value Theorem on [, ]. 7. I '( ) then ( ) 8. I on [-, ], then the Riemann Sum or () on the given interval is. 9. Evaluate the ollowing: / a. sin cos b. c. sec tan sec d. e 9 / e.. cosh sinh g. h. i.

5 j. k. 8 e / cos.. Evaluate. Find the area bounded between y 9 and y.. A tree has been transplanted and ater t years o growing at a rate o dh dt years it has reached a height o t. How tall was the tree when it was planted? t t/yr. At two. Sketch a possible graph o a continuous unction y using the graph o ' i. shown below,

6 Evening: 9 satisies the hypotheses o Rolle s Theorem on [, ].. Find the domain o. Veriy that. I ' then.. Is dierentiable or all in [-, ]?. I y. then 6. I, ' 7, and h then ' 7. I then ind. 8. I h. and g then ind g. What is the domain o g? 9. Determine the horizontal asymptotes and vertical asymptotes o y.. Find or the ollowing: a. y ln e b. y sin cos c. y e sin d. y ln e. y cosh. y7 log g. y y h. y. Evaluate the ollowing: a. sin cos 6

7 b. c. d. e 9. Find. Find tan. Find /. Evaluate 9 6. Find the area o the region bounded by y and y 7. Sketch a possible graph o a unction with the ollowing properties. Domain and Range First Derivative Second Derivative ',, " ' ' ' ",, 8. For sin or [, ] a. Find '. Use it to ind critical values o and intervals where and decreasing. " b. Find. Use it to ind inlection points o and intervals where up and concave down. is increasing is concave 9. A circular oil slick o uniorm thickness is caused by a spill o m oil. The thickness o the oil is decreasing at a rate o. cm/h. At what rate is the radius o the slick increasing when the radius is 8 m?. A bo with a square base and no top must have a volume o, cm. Find the dimensions o the bo that minimize the amount o material used. Veriy with the second derivative test. 7