Mth Calculus Practice Eam Questions NOTE: These questions should not be taken as a complete list o possible problems. The are merel intended to be eamples o the diicult level o the regular eam questions. Questions are listed b section, so be sure to onl stud the questions rom the sections our particular eam is covering. This is onl a list o questions use a separate sheet to work out the problems.. (. and.) Use the given graph to answer each question. ( ) a. lim ( ) b. lim ( ) + c. lim ( ) 0 d. ( 0) e. Does the limit o ( ) eist as? Wh or wh not?. Is the graph o continuous at = 0? Wh or wh not?. Find the ollowing limits, i the eist. I the limit is an ininite limit, speci or. Making a table o values is not an acceptable method. 6 a. (.) lim + b. (.) lim c. (.) lim + d. (.) lim 8 5 5 e. (. ) lim+ [ ]. (.) Complete the table and use the result to estimate the limit. + lim 5. 9. 99. 999. 00. 0. ( )
. (.) Find the limit L. Then ind a δ > 0 such that ( ) L < 0. 0 whenever 0 < c < δ. To receive ull credit, ou must show that our δ makes the above statement true. ( ) lim 5. (.) Find ( + ) ( ) lim i ( ) = + 0. 6. (.) Use the Intermediate Value Theorem to show that there is a root to the equation + = 0 0,. in the interval ( ) 7. (.) Let ( ) = [[] ]. (the greatest integer unction o ) Note that ( 0 ) = 0 and ( ) = but there is no so that ( ) =. Wh doesn t the Intermediate Value Theorem appl to? 8. (.5) Find all vertical asmptotes o the unction ( ) =. + 7 + 0 9. (.) Using the deinition o the derivative, ind '( ) i ( ) =. 0. (.) Given the graph o ( ) below, sketch the graph ( ) = () '. = (). (.) I an arrow is shot upward on the moon with a velocit o 58 m/s, its height (in meters) ater t seconds is given b H = 58t 0.8t. At what time will the arrow reach its maimum height?
. (.) Draw two dierent continuous unctions whose derivatives do not eist at =.. Find the derivative. DO NOT SIMPLIFY YOUR ANSWER. a. (.) π = b. (.) g ( ) = ( ) 5 c. (.) = sin( cos ) 5 d. (.) = + 6 + e. (.) ( ) = sec csc sec. (.) =. (.) Find an equation or the tangent line to the curve sin = at the point (, 0) π. d 5. (.5) Find i + = d d 6. (.5) Find d i = 6. (Notice that I am asking or the second derivative.) 7. (.6) A -t ladder resting on horizontal ground is leaning against a vertical wall when its base starts to slide awa rom the wall. B the time the base is t rom the wall, the base is moving at the rate o 5 t/sec. How ast is the top o the ladder sliding down the wall then?
8. (.) Find an critical numbers o the unction ( ) =. + 9. (.) Locate the absolute etrema o the unction ( ) = 6 on the interval [, ] 0. (.) State wh the Mean Value Theorem can be applied to ( ) [, 9] and ind an value(s) c in the open interval ( ). (.) Find an -value(s) where ( ) = ( ) 0. = on the closed interval, 9 such that () () 9 () ' c =. 9 has relative etrema.. (.) Find the open interval(s) on which the unction ( ) = 6 + 5 decreasing.. (. and.) Given the graph o = ( ) is increasing or, draw irst and second derivative charts. (Something like those shown in problem #9.) Use estimates or an critical numbers and inlection point coordinates. = () ' ''. (.) Suppose ou are given that '( ) = 0 and that ''( ) > 0 or all. Does ( ) relative maimum or minimum at =? Wh? have a
+ =. 5 + 5. (.5) Find an horizontal asmptotes o the unction ( ) 6. (.5) Find lim +. You must show our work to receive credit. + 7. (.6) Given the ollowing inormation, sketch the graph o ( ) =. ' '' - '( ) = 0 '() does not eist ''() does not eist ( ) = ( ) = 0 () 8. (.7) Suppose ou have 50 cm o material available to make a bo with a square base and closed top. Find the height, width and length that would give the largest possible volume. You must use techniques learned in this course to justi our answer. 9. (.8) Consider the equation + + = 0. Wh does Newton s Method ail using an initial guess o =? 5
0. (.8) Use Newton s Method to estimate accurate to 8 decimal places.. (.9) Use dierentials to approimate 6.. (.9) Given = 9, ind the dierential d.. (.9) Suppose an oil compan currentl ships oil in 55-gallon clindrical drums with radius in. and height in., but wants to ship in slightl narrower drums o the same height. I the compan decreases the radius b in., use dierentials to estimate the resulting change in the volume. (You ma leave our answer in inches cubed.) d. (.) Find the particular solution to the dierential equation = given that the graph o d,. passes through the point ( ) 5. Find each indeinite integral. a. (.) d b. (.) + d dt c. (.) ( t sin t) 6. (.) Write the ollowing sum using summation notation. You do not need to ind the sum. 5 5 5 + + + + + + 0 = 7. (.) Find a ormula or this sum in terms o n. n i= ( i + ) n 8. (.) Estimate the area o the region between the curve,. Use our () rectangles and right endpoints. [ ] = and the -ais over the interval 9. (.) Use the limit process to ind the area o the region between the graph o the unction = + 0,. and the -ais over the interval [ ] 6
0. (.) Evaluate the integral ( + )d 0 using a Riemann sum.. Find the ollowing integrals. You must show our work to receive credit. a. (.) ( 6 ) + d b. (.5) cos sin d π c. (.5) π d tan d. (. and.5) 0 ( + ) d. (.6) Use the Trapezoid rule to estimate the number o square meters o land in a lot where and are measured in meters, as shown. The land is bounded b a stream and two straight roads that meet at right angles. 0 58 0 6 0 59 0 5 0 50 50 5 60 70 8 80 8 90 00 0 60 0 0 Road Stream Road 0 0 60 80 00 7