Chapter 5. (AB/BC, calculator) A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t / vt e (a) During what time intervals on [0, 4] is the particle traveling to the left? (b) Find the average velocity of the particle from0 t 4. (c) Is the speed increasing or decreasing at time t =? (d) For the time interval 0t 4, what is the farthest distance to the right that the particle travels from the origin?
. (AB/BC, calculator) e Let f be a function defined by f( ) e (a) Find, if any, the -coordinates of all relative minimum values of f(). (b) Find the average value of f () for < <. (c) Find, if any, the -coordinates of all inflection points of f ().
3. (AB/BC, calculator) Water is flooding into a basement at a rate of f () t t 5. Water is being pumped out at a 0.5t rate of gt () 5 e. Both f (t) and g(t) are defined for 0 < t < 5 hours. At time t = 0, there are 5 cubic feet of water in the basement. (a) Calculate the total amount of water that enters the basement from t = 0 to t = hours. (b) How fast is the water level changing at time t = hours? Use proper units. (c) At what time t was the volume of water at a maimum? How much water was in the basement at the time?
4. (AB/BC, calculator) The function f is defined by f( ) 4 for < <. Let g be the function defined by f( ) for 0 g ( ). + for 0 (a) Is g continuous at = 0? Use the definition of continuity to eplain your answer. (b) What is the area of the region bounded by f, the -ais, the y-ais, and the line =? (c) Find the value of gd ( ).
5. Let f () = arccos() and let g() =. Define h() = f (g()). (a) At what values of does h() have a relative maimum? (b) Write but do not evaluate an epression that can determine the area of the region bounded by the graphs of h() and the horizontal line y. 3 (c) Evaluate d f d 3.
6. (AB/BC, non-calculator) Let f ( ) e. (a) Find the critical values of f. Classify each of these values as a relative minimum, a relative maimum, or neither a minimum nor a maimum. Justify your conclusion. (b) Write the equation of the line tangent to the graph of f at the point where. (c) Given a 0 f ( d ) f a, find a.
7. (AB/BC, calculator) Two towers of equal height are spaced 366 feet apart. A cable suspended between the two forms 5 50 50 a catenary whose height above the ground is given by f( ) e e where = 0 is at the point on the ground halfway between the two towers. (a) What is the height of each tower (rounded to the nearest foot)? (Assume the cable is anchored to the tops of the towers.) (b) Evaluate f (00). Eplain what f (00) is in the contet of the problem. (c) What is the average height of the cable? (d) What is the slope of the cable at the rightmost point where the height of the cable is equal to the average height of the cable?
8. (AB/BC, non-calculator) f() f () 3-3 3 5-4 - 4 5 4 - The table above gives values of twice differentiable at all. f and its derivative f ' at selected values of. The function f is (a) Use the values in the table to estimate the value of f (.5). (b) Eplain why there must be a value c for < c <3, for which f() c. (c) Estimate the average value of f () from < < 4 using a trapezoidal sum with 3 subintervals, as indicated in the table. (d) Assuming f is invertible, find the equation of the line tangent to the graph of 5. y f ( ) at
9. (AB/BC, non-calculator) Let f be a differentiable function such that f ( ) = 3, f (3) = 5, f ( ) = 4, f (3) = 6. Let g be a differentiable function, such that g f ( ) ( ) for all. What is the value of g(3)? (a) 3 (b) 4 (c) 5 (d) 6 (e) 4
0. (AB/BC, non-calculator) If cos ln y, what is dy d in terms of? (a) e cos sin (b) e cos sin (c) sin (d) sin (e) cos sin
Question A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t / vt e (a) During what time intervals on [0, 4] is the particle traveling to the left? (b) Find the average velocity of the particle from0 t 4. (c) Is the speed increasing or decreasing at time t =? (d) For the time interval 0t 4, what is the farthest distance to the right that the particle travels from the origin? (a) The particle travels to the left when v(t) < 0. For t 0, this occurs on the interval 0.70 t.879, so the particle is traveling to the left on that interval. : :answer : justification (b) average velocity 4 0 4 vtdt ( ) 0.03 : : limits and integrand : answer (c) v() = 0.849 a() = v() = 0.699 Since a() and v() have opposite signs, the particle is slowing down. : :answer : justification
Question (cont.) (d) The critical numbers of v(t) for t 0 are t = 0.70 and t =.879. (0) (given) 0 4 0.70 (0.70) v( t) dt.433 0.879 (.879) v( t) dt.070 (4) v( t) dt.588 0 : tests at critical numbers 3: : tests at endpoints : answer The maimum distance from the origin is.433 units.
e Let f be a function defined by f( ) Question e (a) Find, if any, the -coordinates of all relative minimum values of f (). (b) Find the average value of f () for < <. (c) Find, if any, the -coordinates of all inflection points of f (). (a) f ( ) e e e e 0 e e 0 : f 3: : answer : justification f () <0 when < 0 and f () > 0 when > 0, so there is a relative minimum at = 0. (b) average value e e 4 d e e e 4 e : limits and integrand 3: : antiderivative : answer
Question (cont.) (c) f f Since e 0 and e 0 for all values of, f( ) e e 0 for all values of. f Thus 0 f concave up everywhere. Therefore, f does not have any inflection points. : f 3: : answer : justification
Question 3 Water is flooding into a basement at a rate of f () t t 5. Water is being pumped out at a 0.5t rate of gt () 5 e. Both f (t) and g(t) are defined for 0 < t < 5 hours. At time t = 0, there are 5 cubic feet of water in the basement. (a) Calculate the total amount of water that enters the basement from t = 0 to t = hours. (b) How fast is the water level changing at time t = hours? Use proper units. (c) At what time t was the volume of water at a maimum? How much water was in the basement at the time? (a) The total amount of water entering the basement is 0 f( t) dt 6.9ft (b) f () g() = 0.989 ft 3 /hr 3 : :integral : answer : epression for net rate 3: : answer : units (c) R(t) = f (t) g(t) > 0 for 0 < t <.457 hrs. R(t) = f (t) g(t ) < 0 for.457 < t < 5 hrs. Therefore, the volume of water is at a maimum when t =.457 hours. : identifies t.457 4 : : argument for a maimum : total amount of water at maimum point Therefore, the total amount of water is.457 0 5 f( t) g( t) dt 7.89 ft 3
Question 4 The function f is defined by f( ) 4 for < <. Let g be the function defined by f( ) for 0 g ( ). + for 0 (a) Is g continuous at = 0? Use the definition of continuity to eplain your answer. (b) What is the area of the region bounded by f, the -ais, the y-ais, and the line =? (c) Find the value of gd ( ). (a) g 00 lim g ( ) 0 4 0 lim g ( ) 0 0 : calculates left and right side limits 3: : g(0) : conclusion with justification 0 0 lim g ( ) lim g ( ) g 0 g is continuous at 0
Question 4 (cont.) (b) 0 f( ) d arcsin 0 :limits 3: : antiderivative : answer arcsin arcsin 0 6 (c) g( ) d f ( ) d d 0 0 0 arcsin 6 0 : antiderivative of f ( ) 3: : antiderviative of : answer
Question 5 Let f () = arccos() and let g() =. Define h() = f (g()). (a) At what values of does h() have a relative maimum? (b) Write but do not evaluate an epression that can determine the area of the region bounded by the graphs of h() and the horizontal line y. 3 (c) Evaluate d f d 3. (a) h( ) 4 h( ) 0 when 0 h changes sign from positive to negative : h 3: : 0 : justification about 0, so h ( ) has a relative maimum at 0. (b) cos ( ) cos 3 3 so,, and = : solves for 3: : limits : integrand Thus, the area is h ( ) d 3
Question 5 (cont.) (c) f( ) d f d 3 f f 3 f 3 4 : f 3: : f 3 : answer
Question 6 Let f ( ) e. (a) Find the critical values of f. Classify each of these values as a relative minimum, a relative maimum, or neither a minimum nor a maimum. Justify your conclusion. (b) Write the equation of the line tangent to the graph of f at the point where. (c) Given (a) a 0 f ( d ) f a, find a. f e 0 e e 0 Since f 0 for < 0 and f 0 0, a relative minimum occurs at 0. for > : f e 4: : 0 : relative minimum : justification (b) f e f () e : : f : f () and equation The equation is y( e) ( e)( ) y ( e )
Question 6 (cont.) (c) a a f ( d ) e d 0 0 e a a e a 0 : integral 3: : equation : answer e a a a e a 0
Question 7 Two towers of equal height are spaced 366 feet apart. A cable suspended between the two forms 5 50 50 a catenary whose height above the ground is given by f( ) e e where = 0 is at the point on the ground halfway between the two towers. (a) What is the height of each tower (rounded to the nearest foot)? (Assume the cable is anchored to the tops of the towers.) (b) Evaluate f (00). Eplain what f (00) is in the contet of the problem. (c) What is the average height of the cable? (d) What is the slope of the cable at the rightmost point where the height of the cable is equal to the average height of the cable? (a) 366 83, so the towers are at 83 and 83. : : coordinates : f (83) f (83) 60, so the towers are 60 feet tall. (b) f (00) = 0.05 vertical feet per horizontal feet f (00) represents the slope of the cable : : f (00) : eplanation
Question 7 (cont.) (c) Average height 83 83 f( ) d 36.5 ft 366 : :limits : integrand and answer (d) f( ) 36.5 06.4 f (06.) 0.9 vertical foot per horizontal foot : equation 3: : finds 06.4 : slope
Question 8 f () f () 3-3 3 5-4 - 4 5 4 - The table above gives values of twice differentiable at all. f and its derivative f ' at selected values of. The function f is (a) Use the values in the table to estimate the value of f (.5). (b) Eplain why there must be a value c for < c <3, for which f( c). (c) Estimate the average value of f () from < < 4 using a trapezoidal sum with 3 subintervals, as indicated in the table. (d) Assuming f is invertible, find the equation of the line tangent to the graph of 5. y f ( ) at (a) f() f() 3 f (.5) 4 : answer
(b) f(3) f() 3 Since f is continuous and differentiable, by the mean value theorem, there must eist some value c, < c <3, such that f (c) =. : : difference quotient : conclusion using MVT (c) Average value = 4 3 f( ) d (3 ( )) ( 5) (5 ) 4 3 : epression for average value 3: : trapezoidal sum : answer (d) f (3) 5 (5) 3 f (5) f f (5) f (3) Tangent Line: f : f (3) 3: :( f )(5) : tangent line equation y3 ( 5)
Questions 9-0 9. b g(3) f f (3) f ( ) 4 cos ln y y e cos 0. a dy sin yd dy ysin d dy e d cos sin