Multiple Choice. Circle the best answer. No work needed. No partial credit available. is continuous.
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1 Multiple Choice. Circle the best answer. No work needed. No partial credit available Evaluate lim + (a (b (c (d 0 (e None of the above.. Evaluate lim (a (b (c (d 0 (e + + None of the above.. Find c so that f( { sin(+ if 0 c if 0 is continuous. (a c (b c (c c (d c (e None of the above. 4. The graph of f(t is shown below. If F ( (a 0 (b 0 f(t dt, for what value of is F ( an absolute maimum on [0, 5]? f(t (c (d t 4 5 (e None of the above. 5. Evaluate ( + d (a 6 ( C (b ( C (c 8 ( C (d 4 ( C (e None of the above.
2 Fill in the Blanks. No work needed. Only possible scores given are 0,, and If f( sin then f ( ((sin + ( (cos 7. The vertical asymptote(s of f( + are. The horizontal asymptote(s of f( are y. 8. ( d (t + tan t dt t (t + tan t( + sec t( t (t + tan t ( t t 9. A function which satisfies y ( 4 and y( 5 is given by y( + 0. Let f and g be differentiable functions such that f(0 f (0 f ( 4 g(0 g (0 g ( 5 If h( g(f( then h (0 5 Etra Work Space.
3 Standard Response Questions. Show all work to receive credit. Please put your final answer in the BOX.. (8+4 points Throughout this problem y is defined implicitly as a function of. (a Find the slope of the tangent line to the curve y + 7 y + 9 at the point (,. yy + 7 y + y (y + 7 (( + (y 4y y y y (b Write the equation of the tangent line to the curve at the point (,. or y ( y +. (0 points Use the definition of the derivative as a limit to calculate f ( for f(. (There will be no credit for other methods. f( + h f( + h ( f( + h f( ( + h ( ( + h ( + h ( h f( + h f( ( + h ( f( + h f( h ( + h ( f( + h f( lim h 0 h ( ( f ( (
4 . ( points Let f( + cos( on the interval [0, π]. (a Find the intervals within [0, π] on which f is increasing and the intervals on which f is decreasing. f ( sin giving us crit points at π/6 and 5π/6. Choosing test points and sketching a number line we get our answer: f is increasing on [0, π/6 (5π/6, π] f is decreasing on (π/6, 5π/6 (b Find the intervals within [0, π] on which f is concave up and the intervals on which f is concave down. f ( cos. giving us possible inflection points at π/ and π/. Choosing test points and sketching a number line we get our answer: f is con up on (π/, π/ f is con down on [0, π/ (π/, π] (c Indicate the y-intercept, any local maes/mins, and inflection points. Based on the info above we can determine: y int : (0, local ma : π/6 local min : 5π/6 inflection points : π/, π/ (d Locate the absolute maimum of f on [0, π]. Need to test the points π/6 and π as our only maimum candidates. f(π/6 π/6 + ( / f(π π + 8 Giving us a clear winner of π as the absolute ma. 4
5 4. ( points Evaluate the following integrals: (a + 4 d d d (u u ( du u 8 [ u ] ( 8 + C C 4 (du 8 d (b ( sec ( tan sec ( d ( sec ( tan ( sec ( sec ( tan d d (u sec ( u ( sec ( tan ( du u sec ( ( tan (du sec ( ( tan ( d [ u ] ( ( + C 4 sec + C (c 8 7 d Consider 7 d (u + 7 u d (u 7 u + 7 u / + 7u / du (du d [ ] u 5/ + 4u/ 5 [ ] ( 7 5/ 4( 7/ + + C 5 Giving us the solution 8 [ (4 5/ 7 d + 4(4/ 5 [ ( + 4(8 ] 5 ] [ ( 5/ 5 ] [ ] + 4(/
6 5. (8 points Use the Fundamental Theorem of Calculus to find the derivative of F ( π sin(t t4 + dt F ( sin( + ( sin( + d ( d ( 4 6. (8 points Estimate d using areas of rectangles of equal width, with heights of the rectangles determined by the height of the curve at left endpoints (Do not simplify. f( / 5/ Giving us the final solution: 4 ( d
7 7. (8 points Use a linear approimation to estimate 6. Hint: Is 6 close to a number whose cube root is well-known? Take f( and a 7 because f(a f(7. The final component we need is f (a ( 7 7 Therefore L( f(a + f (a( a + ( 7 7 L(6 + ( ( ( points Find the area of the region enclosed by the graphs of the equations y + and y +. First lets find where these intersect by solving: ( + ( 0, Now we check to see which is greater in this interval by taking a test point (0 is a good choice < 0 0+ so we get the integral: Area [ [ ( + ( + ] d [ + ] d ] + [ ] + [9 9 9] 7
8 9. ( points The top of a foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of feet per second. How fast is the bottom of the ladder sliding along the ground away from the wall when the bottom of the ladder is 5 feet away from the base of the wall? The information above can be transformed into the following mathematical statements. (t + y(t h(t h(t y (t We can define t 0 to be the time at which the bottom of the ladder is 5 feet away from the base of the wall (aka (t 0 5. It is then our goal to find (t 0 according to the statement of the problem. (t + y(t h(t (t + y(t (t (t + y(ty (t 0 (t (t + y(t( 0 (t 0 (t 0 y(t (t 0 y(t 0 0 (t 0 y(t 0 5 (t 0 ( (since 5 + 8
9 0. ( points A cylinder is inscribed in a right circular cone of height 4 inches and radius (at the base equal to inches. What are the dimensions of such a cylinder that has maimum volume? You MUST verify that you have found the maimum. Hint: Recall the formula for volume of a cylinder: V πr h The main restriction is that we would like the cylinder to push right up against the cone to maimize the volume. Therefore we need the restriction h 4 4 r, giving us the volume equation: V πr h (r is in [0, ] πr (4 4 r ] 4π [r r V 4π [ r r ] Giving us critical points are r 0 and r. Choosing test points for V we can see that V is increasing on (0, and decreasing on (, ]. Our only candidate for the absolute maimum is r. Giving the corresponding h 4/. 9
10 . ( points The graph to the right shows the velocity v(t in meters per second of a particle moving on a horizontal coordinate line, for t in seconds within the closed interval [0, 8] v(t 4 (a When is the particle moving forward? t [0, 4] (b When is the particle s speed decreasing? (f What is the change in the particle s position from t to t 6? + 4 t [, 4] [6, 8] (c When is the particle s acceleration positive? (g What is the total distance the particle travels from t to t 6? t [0, ] [6, 8] (d When is the particle s acceleration the greatest? t (h If the particle is at the origin at t use linear approimation to estimate its position at t / (e When does the particle move at its greatest speed? t 6 L( 0 + ( s(/ 0
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