MA 123 Calculus I Midterm II Practice Exam Answer Key

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1 MA 1 Midterm II Practice Eam Note: Be aware that there may be more than one method to solving any one question. Keep in mind that the beauty in math is that you can often obtain the same answer from more than one approach. For this reason, the answers provided in this key are left in their most simplified form. Unless otherwise noted, the answers do not need to be simplified, yet simplification may be the key to dissecting a seemingly unworkable function. 1) Differentiate the following functions with respect to (a.) f ( ) 1 / f ( ) 1 / ( ) 1 () ( ) 1+ / f // (b.) f ( ) e ( ) ( ) ( 1) 1 / 1 / z f e + () + ( + 1) (e ) 1 / f ( ) e ( ( + 1) 1 / + ( + 1) ) // (c.) f ( ) e (cos + c) ; Where c is some arbitrary constant f ( ) e ( sin + c) + e (cos + c) f '( ) e (cos sin c) + + // c α (d.) f ( ) sin( e ) ; Where α is some arbitrary constant f ( ) αe cos( e α α // ) ) Use Logarithmic Differentiation to find the derivative of the following function y e ln y ln( e ( + 1) ) ln y ln e + ln + ln( ( +1) 1) 10 ; Separate using log rules ; Simplify using log rules 1 ln y + ln + 10 ln( + 1) ; Differentiate both sides y ( ) ; Simplify and solve for y y y y 1+ + ( e ( + 1) ) 1+ + //

2 MA 1 Midterm II Practice Eam ) Use Implicit Differentiation to find the equation of the tangent line to the following curve at the given point. + y y + at the point (1,) + ( y + y 1) yy + 1 ; Differentiate both sides + y + y yy + 1 ; Simplify y yy 1 y y ( y) 1 y 1 y y y ; Determine (1,) 1 ( 1) ( ) 7 y m ( 1) ( ) ; Use Point Slope Formula 7 y ( 1) ; Solve for y 7 y // ) Let f ( ). (a.) Find the first and second derivatives of f (). f ( ) f ( ) 1 // (b.) On which intervals is f () increasing? decreasing? f ( ) ; Find critical points ( 1) ( + 1)( 1), 1, + 1 ; Plot the critical points on a number line f () is increasing on the intervals (-1, 0) and (+1, ) f () is decreasing on the intervals (-, -1) and (0, +1) //

3 MA 1 Midterm II Practice Eam (c.) On which intervals is f () concave upward? concave downward? f ( ) 1 ; Find inflection points 1 1 ± 1 ; Plot the inflection points on a number line 1 1 f () is concave upward on the intervals, and +, 1 f () is concave downward on the interval, (d.) What are the inflection points of f ()? + 1 // + The inflection points are the -values of which f ( ), namely 1 ± // 5) Water is leaking out of an inverted conical tank at a rate of 10,000 cm /min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6 m and the diameter at the top is m. If the water level is rising at a rate of 0 cm/min when the height of the water is m, find the rate at which water is being pumped into the tank. (Hint: See Figure on Page 6 for a better visualization) If C the rate at which water is pumped in, then dv dt C 10,000, where V ⅓πr h. By similar triangles, r h h r 6 V ⅓πr h ⅓π(h/) h (π/7)h

4 MA 1 Midterm II Practice Eam dv π dh h. When h 0 cm dt 9 dt π 00 dt 9 dh cm/min, so C 10,000 ( ) (0 ) C 10, ,000 π // 9 6) Let f ( ). (a.) (b.) Find any/all local ma and local min of f (). ( f ( ) )( ( ) ( ) )() 1 ( ) 1 Critical Points: f ( ) ( ) ( f ( ) ) 1 ( 1), 1, + ( f ( ) 8 f ( ) ( ( )[( + 96 ) ) ( )( ( ( ) ) 1 ) ( 1 )()( 1 )() )()] Since f ( 1) and f ( 1) < 0, then f has a local maimum at 1. Since f ( + 1) and f ( + 1) > 0, then f has a local minimum at + 1. Since f ( 0) and f ( 0), then f (0) is neither a local ma or min. // Determine when f () is increasing. Plot the Critical Points found in part (a.)

5 (c.) (d.) (e.) MA 1 Midterm II Practice Eam f () is increasing on the intervals (, 1) and ( +, + ) Determine when f () is decreasing. See number line constructed in part (b.) above. f () is decreasing on the intervals ( 1, 0) and (, 1) Find all asymptotes of f (). Vertical asymptotes: When f () approaches ± ( )( + ), + Vertical asymptotes eist at ±. 1 // 0 + // Horizontal asymptotes:: What f () approaches as ± lim lim - lim lim Both limits do not eist due to the fact that they do not converge to a finite number, thus are no horizontal asymptotes. // Determine when f () is concave upward f ( ) ; Find inflection points ( ) ( + 1) ; Only real inflection point Must also find when f ( ) is undefined. i.e when ( ) ( ) ( + )( ), + ; Note: These are the vertical asymptotes

6 MA 1 Midterm II Practice Eam Plot the inflection points on a number line (f.) (g.) f () is concave upward on the interval (, 0) and (, + ) Determine when f () is concave downward.. // f () is concave downward on the interval (, ) and (, ) 0. // Use your results from parts A through F to sketch the graph of f () on the graph below. // Midterm II : Thurs, June 1 th at 6 pm in MCS B