Mat 270 Final Exam Review Sheet Fall 2012 (Final on December 13th, 7:10 PM - 9:00 PM in PSH 153)

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1 Mat 70 Final Eam Review Sheet Fall 0 (Final on December th, 7:0 PM - 9:00 PM in PSH 5). Find the slope of the secant line to the graph of y f ( ) between the points f ( b) f ( a) ( a, f ( a)), and ( b, f ( b )). b a. Find the equation of the secant line to the graph of y f ( ) between the points at and. y6. Find the equation of the tangent line to the graph of y f ( ) at the points. y. Find the equation of the normal line to the graph of y f ( ) at the points. y 5. Use conjecture method to evaluate the it f( ) 9 at = Evaluate the following its using algebraic process: a) b) / ( ) 0 c) 6 0 d) e) 5 cos f) tan g) h) i) j) ( / ) e /8 / DNE DNE 9 cos ( / ) / ( ln tan ) / 7. Find vertical and horizontal asymptote(s) if any: VA: 0 and HA: y 0 f( ) 0 9

2 56 8. Find vertical and horizontal asymptote(s) if any: f( ) VA: 0 and horizontal asymptote HA is y. 9. Write the it definition of derivative for the function y f ( ) at = a 0. Use it definition of derivative to find f ( ) of the function f ( ). Find the equation of the tangent line to y at the point (, ). Find the equation of the tangent line to y e at the point ln y ln. Where is the slope of the tangent line to the curve y equal to? At.5. Find the derivative y f ( ) : ) f ( ) 0 e f ( ) 0 e ) f ( ) e ) sin cos f( ) sin sin (cos sin ) cos (sin cos ) f( ) sin ) sin cos f( ) sin cos 5) f ( ) cos sin f( ) 0 6) ln f ( ) (ln ) ln( ) f( ) 7) / f ( ) (sin 9) 8) f ( ) ( ) 9) f ( ) 0) f( ) ) f ) f ( ) ) ( ) cos tan ( ) y 5 / / 5. Find the equation of the tangent line to y g( ) at = where f (), f (), g( ) f ( ) y Find the equation of the tangent line to y at = 9

3 cos 7. Find the equation of the tangent line to y at / cos cos 8. Find the equation of the normal line to y at / cos 9. The slope of y f ( ) at (,7) is 5 0. The function f ( ) is one-to-one for 0. Find, find f (7). 5 ( f ) (). Find ( f ) (), the derivative of the inverse function of f if f ( ) 7.. A cost function of the form C( ) reflects diminishing returns to sale. Find the and graph the cost, average cost and marginal cost functions.. An 8 foot ladder is leaning against a wall. How fast is the top of the ladder sliding down the wall if the bottom of the ladder is sliding directly away from the wall at ft/sec and the foot of the ladder is ft away from the wall? / 5. A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. Given the surface area S r. Hint: Volume of a dv dr dv sphere is V r r and k r where k is a constant. Now find dt dt dt that dr k dt, shows that rate of change of radius is a constant. 5. Baseball runners stand at first and second base in a baseball game. At the moment a ball is hit, the runner at first base runs to second base at 8ft/sec; simultaneously on the second base runs to the third base at 0ft/sec. How fast is the distance between the runners changing second after the ball is hit? The distance between consecutive bases is 90 ft and the bases lie at the corner of a square. Hints: Draw a rough diagram and choose dd d dy D (90 ) y, find?, given 8, 0. distance is decreasing dt dt dt at a rate of.99 ft/s. 6. Find all relative/absolute ma/min, critical point(s), interval of increase/decrease, inflection point(s) interval of concavity: ) ) ) ) 5) f ( ) ( / 5 ) f ( ) 0 ln 6 f ( ) 0 ln(6 ) f ( ) ( /5 ), [0,] f / ( ) ( ), [,]

4 7. Determine the open interval where the function f ( ) 6 ln( ) is concave up. 8. An 8 ft tall fence runs parallel to the side of a house ft away. What is the length of the shortest ladder that clears the fence and reaches the house? Assume that the vertical wall of the house and the horizontal ground have infinite etend. 5 ft. (see eample at page number 60) 9. Find the dimensions of the right circular cylinder of maimum volume that can be placed inside the sphere of radius R. 0. A 5 cubic foot tank with a square base and an open top is to be constructed of a sheet of steel of a given thickness. Find the length of a side of the square base of the tank with minimum surface area. Length of side of square base is ft.. Linear Approimation: ) Approimate.0 using the linear approimation of f ( ) ( ) m at = 0 m 0.0 ( ) m,.0.05 ) Approimate.0 using the linear approimation of f ( ) at = , (.0).05. Verify the conditions of Mean Value Theorem or Rolle s Theorem and find the value of c in the open interval given. ) ) f ( ), [,] f ( ) ( ), [0,]. Evaluate the following integrals: ) ) (0 sin cos ) e d (0 sin cos ) e d ) a) d b) d c) d d) ) 9 d 5. (cos( ) sin( / )) d 6. sec d 7. a b d tan 8. d sec 9. d 0. sin 0 d e) d

5 sin d. d. d 6. cos d 8. d d sin d (sin cos ) d 9. tan d ln sec C 0. sec d ln csc cot C. csc d cos( 0 ) d sin (cos ) 0 d 0 6 d d d. Evaluate the following: e d ln d 0 sin( ) C 0 (cos ) C ) ) ) ) d d t e dt 0 e t dt d d d t dt d e e d t dt d Given f ( t) t, f (0) 0, find f( ) 6. Find LHS, RHS and Midpoint sum (MPS) use Riemann sum: ) f n ( ), [,],

6 ) f ( ) cos, [,], n ) f ( ), [,], n 6 ) f ( ) cos, [0, / ], n 6 7. Evaluate the sum: 8. Evaluate the sum: 0 n ( n n 0. ) n 0 n [( n ) 0. ] n 9. If two resistors R and R are connected in parallel, as shown in the figure, then the total resistance R measured in ohms (Ω), is given by. R R R If R and R are increasing at a rate of 0. Ω/s and 0. Ω/s respectively, how fast is R changing when R 80 and R 00? 0. Two people start from the same point. One walks east at miles/hour and the other walks northwest at miles/hour. How fast is the distance between the people changing after 5 minutes?. The minute hand on a watch is 8 mm long and the hour hand is mm long. How fast is the distance between the tips of the hands changing at one o clock?. Find (in two decimal places) the coordinates of the point on the curve y tan that is closest to the point (, ).