ACTM Regional Calculus Competition 2018

Transcription

1 ACTM Regional Calculus Competition 08 Work the multiple-choice questions first, choosing the single best response from the choices available. Indicate your answer here and on your answer sheet. Then attempt the tie-breaker questions at the end starting with tie breaker #, then #, and finally #. Turn in your answer sheet and the tie breaker pages when you are finished. You may keep the pages with the multiple-choice questions. Figures aren t necessarily drawn to scale. Angles are given in radians unless otherwise stated... æ4x -6ö limç = 6. By the definition of a limit, there is a positive real number d such that x è x - ø 4x -6-6 < 0.4 if 0< x - < d. The largest valid value of d is x - A. 0.0 B C. 0. D. 0. E. 0.5 æ 4x 5x ö ç + = ( x) ( x) è ø A. Undefined B. C. D. 4 E. 9 lim x 0 sin cos. Which of the following indicates the presence of a horizontal asymptote for the graph of y = f(x)? lim f x = A. x 4 ( ) B. f ( x) x C. lim f ( x) x D. lim f ( x) lim = x = = 4. There is a stack of newspapers whose weight is given by w(t) where t is time. A match is thrown in the stack and we notice that the fire is increasing in vigor at time t =. Which of the following must be true? A. w'(t) > 0 and w"(t) > 0 B. w'(t) > 0 and w"(t) < 0 C. w'(t) < 0 and w"(t) > 0 D. w'(t) < 0 and w"(t) < 0 E. One cannot determine the signs of these derivatives. Page

2 5. ( + ) - '( ) f ' x h f x lim = h 0 h A. Does not exist B. 0 C. f (x) D. f '(x) E. f "(x) ACTM Regional Calculus Competition 08 ( ( )) 6. d dx ln sin ( x ) = A. ln ( cos( x) ) B. cos( ln ( x) ) C. ( x ) cos( x) D. sin ( x ) E. cot ( x) 7. d ( 4 x 5x dx x ) + + = x ( x) A. ln + x + 5 B. - + x + 5 C x x D. x+ 4 x + 5 = + x for 8. The depth of the water x feet from the end of a swimming pool is given by ( ) x [0,0]. What is the average depth of the water on this interval to the nearest tenth of a foot? A..7 B. 4 C. 4. D A region is bounded by the curves x =, x = 4, y = x 4, and y = 4 x. Compute the area of the region. Round your answer to two decimal places. A B. 5.8 C. 5.5 D hx 80 Page

3 ACTM Regional Calculus Competition 08 For problems 0 and. Following is a table of velocities and times since midnight for a vehicle. t hours v miles/hour Give the best estimate of the instantaneous acceleration exactly at 7:00 am. A. mi/h B..75 mi/h C..5 mi/h D mi/h E. 55 mi/h. Use the midpoint rule with 4 intervals to approximate the total distance traveled from :00 am to 5:00 pm. A. 860 mi B. 88 mi C. 40 mi D. 0 mi E. 80 mi. What is the equation of the tangent line to f ( x) A. y = 7 x B. y = x + 5 C. y = 5 x D. y = x + 5 x + = x - at x =? ( ) d. sin ( ) dx x = A. sin( x ) B. sin( x ) C. sin( x ) D. cos ( x ) Page

4 ACTM Regional Calculus Competition 08 For problems 4 and 5. Following is the flow rate of a pollutant in a lake in liters/hour as a function of time in hours. 4. Estimate the total amount of the pollutant which entered the lake from time t = to t = 5. A. 5 liters B. 0 liters C. 5 liters D. 0 liters E. 0 liters 5. How fast the flow rate changing at t = 4.4? A. Decreasing at.5 liters/hour/hour B. Decreasing at.5 liters/hour/hour C. Decreasing at.5 liters/hour/hour D. Increasing at.5 liters/hour/hour E. Increasing at.5 liters/hour/hour d 6. ( sec( x) tan ( x dx )) = A. sec ( x) - sec( x) B. sec( x ) C. sec ( x) tan ( x ) D. cos ( x) - sin ( x) cos ( x) Page 4

5 ACTM Regional Calculus Competition If p() = 4, p'() = 0 and p"() = which of the following must be true about the graph of s(x)? A. The graph has a local maximum at (, 4). B. The graph has a local minimum at (, 4). C. The graph has an inflection point at (, 4). D. There is a hole in the graph at (, 4). 8. sin( 5x d ) ( ) dx e = A. 5 cos( 5 ) B. 5 x x e ( x ) sin 5 xe ( x ) cos 5 xe ( x ) sin 5 C. 5 D. cos( 5x ) e 9. A container is in the shape of a square pyramid with the vertex at the bottom. Its base is meters on each side, and its height is 4 meters. It is being filled with water at a rate of 9 m /min. How fast is the depth of the water growing when the depth is meters? The volume of a pyramid is given by V = # $ Bh. A. meters/minute B. meter/minute C. 0.5 meters/minute D. 0.5 meters/minute 0. There is a line going from the origin to a point on the graph of what is the slope of the one with the largest slope? A. e B. C. D. 9e - x y= x e, x³ 0. Of all such lines, Page 5

6 ACTM Regional Calculus Competition 08. The region R bounded by the graphs of y= x, y =, and x = 4 is revolved around the y-axis to form a solid of revolution. The volume of this solid is given by the integral 4 9 A. 4 pò æ ç x - ö dx è ø 4 B. p ò ( x -) dx 6 C. p ò æ ç x - ö dx è ø D. p ò æ ö ç - y dy è ø 8 4 E. pò æ ç 6 - y ö dy è ø. Here is a table of values for a function y= f() x : x f( x ) The values in this table suggest A. B C D E. The limit does not exist. lim f( x) = x 7 x- 7. lim æ x + ö ç = x x è x - ø A. - B C. 0 D. 5 E. Page 6

7 æsin( x) - sin( a) ö 4. lim x a ç = è x- a ø A. - cos( a) B. sin ( ) C. 0 D. cos( a ) E. None of the other answers is correct. ACTM Regional Calculus Competition The function gx () has a derivative '( ) b ò g '( x) dx can be interpreted as a g x that is continuous over an interval [, ] A. The net area between the graph of y= g( x) and the x-axis between x = a and x = b. B. The average rate of change of y= g( x) between x = a and x = b. C. The average rate of change of y= g'( x) between x = a and x = b. D. The net change in the function y= g( x) between x = a and x = b. E. The net change in the function y= g'( x) between x = a and x = b. ab. The definite integral Page 7

8 ACTM Regional Calculus Competition 08 Tiebreaker Question Name School Let f(x) be a differentiable function and c be a constant real number. Let g x = c f x. Complete the following statement: g, x =. Prove this result. Page 8

9 ACTM Regional Calculus Competition 08 Tiebreaker Question Name School ax Consider the family of functions of the form f( x) =, where a, b, and c are all non-zero real numbers. bx + c Answer the following in terms of a, b, and c. Justify all answers. a) Identify any discontinuities of the function. Determine whether the discontinuities are removable or non-removable. b) Find f '( x ). c) Find f ''( x ) d) Determine the intervals where f( x ) is increasing or decreasing. Page 9

10 ACTM Regional Calculus Competition 08 Tiebreaker Question Name School The following table gives various values of a function and its derivatives. x f( x ) f '( x ) f ''( x ) Furthermore, f ''( x ) is continuous for all real numbers x. Is it possible for the line x = to be a vertical asymptote for the graph y= f() x? Explain. Page 0

11 ACTM Regional Calculus Competition 08 Solutions C C B 4 D 5 E 6 E 7 C 8 D 9 B 0 B A A B 4 E 5 C 6 A 7 B 8 A 9 D 0 A E E B 4 D 5 D Page

12 ACTM Regional Calculus Competition 08 Tiebreaker Question Solution Let f(x) be a differentiable function and c be a constant real number. Let g(x) = c f(x). Complete the following statement: g'(x) = c f '(x). Prove this result. Proof: g' ( x) ( + ) - ( ) g x h g x = lim h 0 h cf x h cf x = lim h 0 h c f x h f x = lim h 0 h f ( x+ h) - f ( x) = c lim h 0 h = cf' ( + ) - ( ) ( ( + ) - ( )) ( x) Definition of Derivative Definition of g(x) Distributive Property [ca + cb = c(a + b)] Limit Property: élim( cp( h) ) = clim( p( h) ) ù ëh a h a û Definition of Derivative Page

13 Tiebreaker Question Solution ACTM Regional Calculus Competition 08 ax Consider the family of functions of the form f( x) =, where a, b, and c are all non-zero real numbers. bx + c Answer the following in terms of a, b, and c. Justify all answers. a) Identify any discontinuities of the function. Determine whether the discontinuities are removable or non-removable. b) Find f '( x ). c) Find f ''( x ) d) Determine the intervals where f( x ) is increasing or decreasing. a) Since f is a rational function, it is continuous throughout the domain. The only discontinuity is at the c zero of the denominator, x =- b. At this value of x, the numerator will be ac - ¹ b 0 since a and c are both not zero. Thus, lim f( x) does not exist, so f has a non-removable discontinuity. b x - c ( ) ( ) ( ) ( ) bx + c a -ax b ac b) Quotient Rule: f '( x) = = bx + c bx + c Product Rule f '( x) ax( bx c) b a( bx c) ( bx c) abx a( bx c) ) = = + é- ë + + ù= û or ac ( bx + c) d - - abc f x = ac bx+ c =- ac bx+ c b=- dx ë û bx + c c) ''( ) é ( ) ù ( ) d) The denominator ( ) ( ) bx + c of the derivative f '( x ) will be positive for all real values of x with c x ¹-. Thus the sign of f '( x ) will depend on the sign of the product ac. b If a and c are both positive or both negative, then f '( x ) will be positive and thus f( x) will be æ c ö increasing on the intervals ç -,- è b ø and æ b c, ö ç - è ø. If a and c have different signs, then f '( x ) will be negative and thus f( x) will be decreasing on the æ c ö intervals ç -,- è b ø and æ b c, ö ç - è ø. Page

14 ACTM Regional Calculus Competition 08 Tiebreaker Question Solution The following table gives various values of a function and its derivatives. x f( x ) f '( x ) f ''( x ) Furthermore, f ''( x ) is continuous for all real numbers x. Is it possible for the line x = to be a vertical asymptote for the graph y= f() x? Explain. No. The continuity of f '' implies that f ' is differentiable, and thus continuous, for all x. Repeating the logic gives f is continuous for all x. Since f is continuous for all x, there can be no vertical asymptote. Page 4