Key- Math 231 Final Exam Review
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1 Key- Math Final Eam Review Find the equation of the line tangent to the curve y y at the point (, ) y-=(-/)(-) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y y (ysiny+y)/(-siny-y^-^) Find the eact area bounded by y ( ), the -ais, = -, and = Find the eact area bounded by, f ( ), and the -ais / + pi/ tan 6 Find lim ln( ) pi 7 Find lim e cos Find y if y (tan ) y' y( sin ln(tan ) csc ) (tan ) cos ( sin ln(tan ) csc ) 9 Find y if y (cot ) y (cot ) ( ln(cot ) / cos sin ) Given g ( ) sin( t ) dt, find g () g' ( ) sin( )
2 A particle moves along a line with velocity function v( t) t t, where v is measured in meters per second Find (a) the displacement and (b) the distance traveled by the particle during the time interval [,] 7/6 Find g () if g csc ( ) 6-6^{sqrt{csc }} ln6 (/)(csc )^{-/}csccot 9 For a function g(), given that g ( ) and g ( ), find all intervals where g() is simultaneously decreasing AND concave down (-infty, ), (/,9/) Consider the function g() whose graph is shown below Find each of the following If the limit does not eist, describe the outcome as,, or LDNE y a) lim g( ) b) lim g( ) pos_inf c) lim g( ) d) lim g( ) Compute the EXACT value of each limit, if it eists If the limit does not eist, describe the outcome as,, or LDNE a) lim -inf 7 b) lim 6 (-sqrt7)/6
3 c) lim f ( ) 7 if f ( ),, d) lim e) lim 6 6 / f) lim ( ) inf g) lim 6 7 h) 7 lim (sqrt7)/ 6 a) Find the -coordinate(s) of all point(s), if any, on the graph of the curve none which the tangent line is parallel to the line y b) Find the equation of the normal line to the curve y-/=9(-) y at = y at 7 Use the limit definition of the derivative to find for the function f ( ) Use the limit definition of the derivative to find for the function f ( ) 9 a) Epress the definite integral e d as the limit of a Riemann sum b) Evaluate the Riemann sum for points to be the left endpoints f ( ) e, [,] with subintervals taking the sample
4 a) e d lim e n n i ( i ) /6 9/6 b) e e e 9 / e 9 9 Find the EXACT value of c so that the function f given below is continuous at = 9 9 (9 )(9 ) f ( ) c c 7 for 9 for 9 State the three points to be satisfied for a function f () to be continuous at = a f(a) eists limit as approaches a eists above two are equal Use the above definition to determine whether discontinuous at f ( ) is continuous or a) = cont b) = hole c) = inf d) If the function is discontinuous at =, and/or, describe the type of discontinuity Sketch the graph of a function f () that satisfies the following conditions f ( ), f ( ), f() has eactly one point of discontinuity f ( ) f ( ) on (,), f ( ) on (,) (, ) f ( ) on (,) (, ), f ( ) on (, ) lim f ( ), lim f ( )
5 y - Given the function f ( ), a) Find the linear approimation, L(), to the function at a = b) Use your result from part (a) to approimate Answer to decimal places or eact value a) L ( ) ( ) 6 b) L ( ) Given the function f ( ) sin( ), a) Find the linear approimation, L(), to the function at a = b) Use your result from part (a) to approimate sin( ) a) L ( ) b) L ( 9) A woman standing on a cliff is watching a motorboat through a telescope as the boat approaches the shoreline directly below her If the telescope is feet above the water level and if the boat is approaching at feet per second, at what rate is the angle, A, of the telescope changing when the boat is feet from the shore? (radian mode)
6 da dt rad sec rad sec 7 6 A long rectangular sheet of metal, 9 inches wide, is to be turned up at both sides to make a horizontal gutter with vertical sides How many inches should be turned up at each side for maimum carrying capacity? 9 > > in 7 Using calculus, determine the absolute minimum and maimum for the function 6 f ( ) on the interval, absol min of -, absol ma of -/ Using calculus, determine all local minimum and maimum for the function f ( ) e ( ) (change problem-find values only) ma at -/ (sqrt)/, min at -/ + (sqrt)/ 9 Suppose a function g() has the following derivative: On what interval(s) is g () increasing? (-inf,-7/),(,inf) g ( ) ( )( 7)( ) Use Newton s method to find all roots of the given equation to 6 decimal places cos 67 An object was dropped off a building and hit the ground with a speed of 7 feet per second ft What is the height of the building? (use a( t) ) 9/6, which is approimately equal to 66 s
7 Let s ( t) t t, t be a position function for an object that moves along a horizontal line Chart the motion of the object When is the object speeding up? When is the object moving right? the object is speeding up when t > The object is also moving right when t > Evaluate the Riemann sum for f ( ) ln( ), [,] with subintervals taking the sample points to be the right endpoints 677 a) What integral does the Riemann sum approimate? b) Epress the definite integral in part a) as the limit of a Riemann sum Evaluate each integral ( )( ) a) d 6 b) d ln-/ pi/ c) ( e cos sin cos9) d -e^cos+/9*sin9+c d) (csc (ln ) e) d sin ) d -/*cot + ((arcsin)^)/ +c ((ln)^)/ +c f) d (/6)(6^9-^9) Find the interval on which y dt t t is concave up The function is concave up when < -/
8 6 Let g( ) f ( t) dt where f is the function whose graph is shown At what values of do the local maimum and minimum values of g occur? f(t) t Local ma at =,6 Local min at = 7 Find all values of c in the given interval that satisfy the Mean Value Theorem for : f ( ), [,] +- /sqrt Let f ( ), f ( ) 6 for How large can f () possibly be? 6 9 Graph f ) Label all asymptotes, local ma/mins, inflection points, if they eist 9 ( The diameter of a sphere is measured to be 6 inches with a possible error of inches Use differentials to estimate the maimum error in the calculated volume 6pi() This review may not cover every topic on the final eam It is essential to reread the tet, go over your instructor s class notes, review all assigned HW then find similar problems in the chapter reviews for more drill Begin your final eam review at least weeks prior to the final
Math 231 Final Exam Review
Math Final Eam Review Find the equation of the line tangent to the curve 4y y at the point (, ) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y 4 y 4 Find the eact
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