Math 231 Final Exam Review

Transcription

1 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 area bounded by y ( 4), the -ais, = -, and = 5 Find the eact area bounded by, and the -ais, 6 Find tan ln( ) 7 Find e Find y if y cos (tan ) 9 Find y if y (cot ) 5 Given g ( ) sin( t 4 ) dt, find g () 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 [,5] Find g () if g csc ( ) For a function g(), given that g ( ) and g ( ), find all intervals 4 where g() is simultaneously decreasing AND concave down

2 4 Consider the function g() whose graph is shown below Find each of the following If the it does not eist, describe the outcome as,, or LDNE 4 y a) g( ) b) g( ) c) g( ) d) g( ) 5 Compute the EXACT value of each it, if it eists If the it does not eist, describe the outcome as,, or LDNE a) b) 5 6 c) if 5,, d) e) 6 6 f) ( ) g) h) 7 5 4

3 6 a) Find the -coordinate(s) of all point(s), if any, on the graph of the curve y which the tangent line is parallel to the line y at b) Find the equation of the normal line to the curve y at = 5 7 Use the it definition of the derivative to find for the function Use the it definition of the derivative to find for the function 9 a) Epress the definite integral e d as the it of a Riemann sum b) Evaluate the Riemann sum for points to be the left endpoints e, [,] with 4 subintervals taking the sample Find the EXACT value of c so that the function f given below is continuous at = 9 9 (9 )(9 ) c for 9 for 9 State the three points to be satisfied for a function f () to be continuous at = a Use the above definition to determine whether discontinuous at is continuous or a) = b) = c) = d) If the function is discontinuous at =, and/or, describe the type of discontinuity

4 Sketch the graph of a function f () that satisfies the following conditions f ( ), f ( ), f() has eactly one point of discontinuity f ( ) f ( ) on (,4), f ( ) on (,) (4, ) f ( ) on (,4) (4, ), f ( ) on (, ), 4 y - 5 Given the function, a) Find the linear approimation, L(), to the function at a = b) Use your result from part (a) to approimate 5 Answer to 4 decimal places or eact value 4 Given the function sin( ), a) Find the linear approimation, L(), to the function at a = b) Use your result from part (a) to approimate sin( ) 5 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 5 feet per second, at what rate is the angle, A, of the telescope changing when the boat is 5 feet from the shore? (radian mode)

5 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 > > 7 Using calculus, determine the absolute minimum and maimum for the function 6 5 on the interval, Using calculus, determine all local minimum and maimum for the function e ( ) 9 Suppose a function g() has the following derivative: On what interval(s) is g () increasing? 4 g ( ) ( )( 7)( 5) Use Newton s method to find all roots of the given equation to 6 decimal places cos 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) ) 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? Evaluate the Riemann sum for ln( ), [,] with 4 subintervals taking the sample points to be the right endpoints a) What integral does the Riemann sum approimate? b) Epress the definite integral in part a) as the it of a Riemann sum s

6 4 Evaluate each integral ( )( ) a) d 6 b) d c) ( e cos sin cos9) d d) (csc (ln ) e) 4 d sin 4 ) d f) 5 d 5 Find the interval on which y dt t t is concave up 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

7 7 Find all values of c in the given interval that satisfy the Mean Value Theorem for :, [,] Let f ( ), f ( ) 6 for 5 How large can f (5) possibly be? 9 Graph f ) Label all asymptotes, local ma/mins, inflection points, if they eist 9 ( 4 The diameter of a sphere is measured to be 6 inches with a possible error of 5 inches Use differentials to estimate the maimum error in the calculated volume 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