8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure.. Write a primar equation. Isolate whatever is to be maimized or minimized.. Rewrite with onl one variable on each side. This ma require a secondar equation. 4. Find the domain. 5. Take the derivative, find critical numbers, make a number line, etc. Eamples: Answer with a complete sentence.. A bo with no lid is to be made from 48 cm of material. If the bo must have a square base, find the dimensions that produce a maimum volume.. The product of two positive numbers is 88. Find the two numbers so that the sum of twice the first plus the second is as small as possible.
8 ASSIGNMENT 5- Write sentence answers on -6.. The product of two positive numbers is. Find the two numbers so that the sum of the numbers is as small as possible.. The area of a rectangle is 8 cm. Find the length and width so that the rectangle has a minimum perimeter.. The perimeter of a rectangle is 8 feet. Find the length and width so that the rectangle has a maimum area. 4. Two adjacent rectangular corrals are to be made using 4 feet of fencing. The fence must etend around the outer perimeter and across the middle as shown in the diagram. Find the dimensions so that the total enclosed area is as large as possible. 5. A shelter at a bus stop is to be made with three Pleiglas sides and a Pleiglas top. If the volume of the shelter is 486 cubic feet, find the dimensions that require the least amount of Pleiglas. 6. A bo is made b cutting small squares from each corner of a piece of square material inches on each side and then folding up the flaps. Find the side of the square cutouts that will produce the greatest volume bo. 7. A rectangle is positioned with one verte on the line as shown. Find the point so that the rectangle has a maimum area., 8. Use the graph of f shown to graph f and a graph of f with the starting point (,). f
8 9. The volume formula for a cone is find dv dt when r = 6 inches. V dr in r h. If and h r, dt min. An airplane fling at an altitude of miles flies directl over a radar station. When the plane is 5 miles awa from the station, the radar shows the distance s is changing at the rate of miles per hour. What is the plane s speed?. Use the graph of f at the right for these problems. a. Find lim f b. Find lim f c. Find lim f 4 d. Find lim f e. Find lim f..... f. List the discontinuities of f. g. Which of these discontinuities are removable?,. h. Find the absolute maimum of f () on i. Find the absolute minimum of f () on,. j. Find f. k. Find f. l. List all -values where f does not eist. m. List all -values at which f has a local minimum. n. List all -values at which f has a local maimum.
84 LESSON 5- MORE MAX/MIN APPLICATIONS Eample: The sum of two nonnegative numbers is. Find both numbers if the sum of twice the first plus the square of the second is a maimum. ASSIGNMENT 5-. The product of two positive numbers is. Find the two numbers so that the sum of the first plus three times the second is as small as possible.. The sum of two nonnegative numbers is 5. Find the two numbers so that the sum of the first plus the square of the second is a minimum.. The sum of two nonnegative numbers is 5. Find the two numbers so that the sum of the first plus the square of the second is a maimum. 4. A rancher plans to fence in three sides of a rectangular pasture with the fourth side being against a rock cliff. He needs to enclose, square meters of pasture. What dimensions would require the least amount of fence material. 5. A bo is to be made b cutting small squares from each corner of a ft b 5 ft rectangular piece of material. Find the size of the square cutouts that would produce a bo with maimum volume. (Your V = equation will not be factorable. You ma use a calculator to solve it.) Show three or more decimal place accurac. 6. Find the volume of the bo in Problem 5. Show or more decimal place accurac.
85 7. A rectangle is positioned with two points on the semicircle 6 as shown. Find the point (,) so that the area of the rectangle is a maimum. 8. Find the area of the rectangle in Problem 7., 9. A bo with an open top has a square base. If the volume of the bo is 4 cubic centimeters, what dimensions minimize the amount of material used?. Find the relative etrema and points of inflection and graph f, if f 4 f 4 6 and 4.. Find the relative etrema and concavit and graph g. Find the derivative... f 4. g 5. If f, find f and graph both f and f. 6. Use a graphing adjustment of a parent graph to graph. 7. Use the graph of f shown to graph f and a f possible graph of f.
86 LESSON 5- APPROXIMATING WITH THE TANGENT LINE In man instances, finding a value of a function is difficult or impossible. With the use of Calculus techniques, we can approimate the function value b finding a -value on a tangent line to the function. Since this method involves using a linear function (the tangent line function) at a nearb point, it is sometimes called a local linearization approimation. Eamples:. If, is a point on the graph of 4, use the equation of a tangent line passing through the point, to approimate a -coordinate (a) when the -coordinate is.. (b) when the -coordinate is.9.. If f and f, use local linearization to approimate. f, ASSIGNMENT 5-. a. Write an equation of the tangent line shown. b. Use this tangent line equation to f.. approimate c. What is the actual value of f.? f 4,
87. Make a large cop of the graph on our own paper. a. Draw the tangent line at the point (,). b. Write an equation of this tangent line. c. Label a point on our tangent line with an -coordinate of.9 as point A. d. Use our equation of the tangent line to approimate f (.9) b finding the -coordinate of our point A.. e. Label a point B on the parabola with an -coordinate of.9. What is the actual value of f (.9)? f. Use the same tangent line to approimate f (.6). How accurate is our approimation? (,) f. Approimate 6 using the equation of a tangent line. You must choose our own equation and point. The graph shown should help. (5,5) 4. The graph of a function f is shown. If f 9, use local linearization to approimate f (.). (5,5) 5. Find the actual value of f (.) from problem 4 or eplain wh it cannot be found. (,) 6. The point (5,) is on the curve the -coordinate when 5 5. 9. Use a tangent line to approimate 7. The length of one side of a square is found to be 8 inches with a possible measurement error of 6 inch. a. Instead of using the actual area formula ( A s ), approimate the area of the square using a local linearization of the area formula if the length of the side is reall 8 inches (without using a calculator). 6 5 b. Find the approimate area if the side is actuall 7 inches. 6 c. Use our answers from parts a and b to give an approimate range of values for the area of the square. 8. Use a tangent line equation to approimate f (8.) if f (without using a calculator).
88 4 9. The point (,) is on the graph of 9. Use the equation of a tangent line to approimate a -coordinate when =... Use a calculator to find an actual -coordinate on the graph of the curve from problem 9 when =.. Show the equation ou are solving.. Given the function a. use the equation of a tangent line to approimate 6 without using a calculator. b. find the actual value of 6?. Find the absolute etrema of the function f on the interval. The second of two positive numbers is the reciprocal of the first. Find the two numbers so that their sum is a minimum. 4. The function f.5,. 4 can be used to model how a disease spreads in an e isolated population of 4 people. represents the time in das since the sickness started and f () represents the number of people who have become sick. Use a calculator to help answer the questions below. a. How man people have become sick b the tenth da? b. How fast was the disease spreading on the tenth da? c. Find the maimum point on f. What does the -coordinate represent? What does the -coordinate represent? Note: Finding a maimum or minimum with a calculator is not allowed on the AP Calculus test. d. How man people have caught the disease when the curve is the steepest? e. Wh would the slope of the curve decrease after a period of time? f. When was the rate of the spread of the disease increasing the fastest? 5. A rancher plans to fence in three sides of a rectangular pasture with the fourth side being against a rock cliff. If he has ards of fencing to use, what is the maimum area he can enclose? 6. Without using a calculator, find vertical asmptotes, relative etrema, and end behavior, and then sketch a graph of f. 9 7. Use graphing adjustments of a parent graph to graph.
89 8. Use the intercepts, vertical asmptotes, relative etrema, and end behavior to graph f, if f and f 4. Do not use a calculator. 44 9. Find all points of inflection of 4 4 f 6.. Find all relative etrema points on the graph of. 4. Find 5 6 lim.. Find 5 4 6 lim.. Find the c-value guaranteed b the Mean Value Theorem for the function 5,. You ma use a calculator. on the interval
9 LESSON 5-4 ANTIDIFFERENTIATION, INDEFINITE INTEGRALS Warm-up Eamples: Differentiate each of the following.. f ( ). f ( ). f ( ) C where C is an constant (number) So what should ou get when ou antidifferentiate? ( ) f ( ) f This problem can be written as d The smbol is called an integral smbol and tells ou to integrate (antidifferentiate) the epression which follows it. That epression is called an integrand. d indicates that ou are integrating with respect to the variable but does not affect the integration process. C is called the constant of integration and must be written as part of our answer when ou are antidifferentiating. Integration Rules: Power Rule: n n d C, n n Constant Rule: If k is an constant, k d k C Scalar Multiple Rule: If k is an constant, k f ( ) d k f ( ) d Sum Rule: (Constants ma be factored out of the integral epression. NEVER factor out a variable.) f ( ) g( ) d f ( ) d g( ) d Eamples: Evaluate (Integrate). 4. d 5. d 6. 4 ( t ) dt
9 7. ( 4 ) d 8. d 9. d Note: Put C when ou integrate, but never when ou differentiate. Sometimes an initial condition is given which makes it possible to solve for C. Eample : If f ( ) and f (), find f ( ). d Eample : Evaluate 5 d d If we know the acceleration equation for an object, and if we are given initial conditions for the object s velocit and position, integration allows us to find the velocit and position equations for the object. Remember: Pos. Vel. Acc. (Differentiate), so Acc. Vel. Pos. (Integrate). Eample : The acceleration of a particle at time t is given b a( t) 4t. v() 6 and s() 5. a. Find the velocit equation. vt () b. Find the position equation. st ()
9 Eample : Given that on earth, the acceleration of an object due to gravit is approimatel ft / sec (negative indicates downward), develop a. the equation for the velocit of the object. vo initial velocit vt b. the equation for the position of the object. so initial position st Note: The two equations vt t v and 6 for an motion affected onl b the earth s gravit. s t t v t s ma be used ASSIGNMENT 5-4 For Problems -4, rewrite the integrand and then integrate.. d. 4 t dt. ( )( ) d 4. Evaluate (integrate) each integral in Problems 5-. 5. ( ) d 6. d 7. ( ) d d 4 9. 5 d. (t ) dt 5 8. d. 4 8 d. t dt t. d 4. If f ( ) 4 and f (), find f ( ).
9 d 5. The derivative of a function is. If the graph of the function contains dt t the point (, ), find the equation of the function. 6. a. Find an equation for the famil of functions whose derivative is. b. Find the particular function from the famil in Part a. whose curve passes through the point (4, ). 7. Find g, ( ) given that: g ( ), g() 5, and g( ). 8. Evaluate d d ( ) d. Hint: This is a derivative of an integral. 9. The acceleration of an object moving along a horizontal path is given b the equation a( t) 6t 4. The object s initial velocit is 5, and its initial position is. a. Find a velocit equation for the object. b. Find the velocit of the object when t. c. Find a position equation for the object. d. Find the object s position when t.. The velocit of an object moving along a vertical path is given b the equation v( t) t, t. a. Find an equation for the object s acceleration. b. Find the acceleration of the object when t 9. c. The object s position at t 9 is. Find an equation for the object s position.. A ball is dropped from a bridge which is 6 feet above a river. How long will it take the ball to hit the water? Use the equation s( t) 6t vot so.. For the first 4 seconds of a race, a sprinter accelerates at a rate of meters per second per second ( m/sec ). He then continues to run at the constant speed that he has attained for the rest of the race. a. Write a piecewise function to epress the sprinter s velocit vt () as a function of time. b. Find v(), v(4), and v (6). c. Write a piecewise function to epress the sprinter s position st () as a function of time. d. How far does the sprinter run during the first 4 seconds of the race? e. How long will it take the sprinter to run m?
94 For Problems and 4, the graph of the derivative ( f ) of a function is given. Sketch a possible graph of the function f.. f 4. f f contains the point (, ) 5. List the domain, vertical asmptote(s), hole(s), - and -intercepts, end behavior, and tpe(s) of smmetr for the graph of. Then sketch the graph without using a calculator. 6. If f f ( ), find ( ). 7. Find an equation of a line tangent to the curve 6 5. which is parallel to the line 8. Find the cubic function of the form a b c d which has a relative maimum point at (, ) and a point of inflection at (, ).
95 LESSON 5-5 THE GENERAL POWER RULE FOR INTEGRALS AND U-SUBSTITUTION In Lesson -, ou learned to differentiate composite functions b using the General Power Rule for Derivatives (Chain Rule for power functions). u is a function of ). We reverse this process when integrating. d (where u n nu n u d General Power Rule for Integrals: (Informall called the Reverse Chain Rule) n n u u ud C, n n This looks a lot like the simple power rule for integration that ou learned in the last lesson. However, the general power rule requires a hook-on factor u to be present before ou can integrate. It is a crucial part of the Reverse Chain Rule. Eamples:. Differentiate ( 5 ) 4. Now, integrate 5( 5 ) Note: You hooked on the derivative of the inside of the power function in Eample, so ou had to unhook the derivative of the inside in Eample. Eamples: Integrate.. ( ) d 4. ( )( ) t t t t dt 5. 6 d 6. 4 5 ( ) d
96 u-substitution For more complicated integration problems, simple rules for integration might fail, and ou ma have to make some tpe of substitution to be able to integrate. In this course, a common substitution will be to let u = the radicand radicand part of the epression and to change the variable throughout the integral before integrating. You should use this method of substitution (called u-substitution) onl when simpler methods don t work. It should be our last resort. Procedure for u-substitution: (for d problems requiring the method). Let u radicand (part inside the smbol).. Solve for (in terms of u).. Differentiate the equation from Step. 4. Find d. 5. Substitute u-epressions for -epressions in the integral. Note: Most often, d du. Don t forget to substitute for d. 6. Integrate. 7. Substitute back, so that our final answer is again in terms of. Sometimes it is easier to do Step before Step. These two steps are reversible. Eamples: Integrate. 7. d 8. d You now have three strategies for integrating.. Term b term using the rules from page 9.. General Power Rule (Reverse Chain Rule).. u-substitution.
97 ASSIGNMENT 5-5 Evaluate (integrate) in Problems -4.. 4. 5 ( ) d. d 5. 8 (5t ) dt. 5 ( ) d 6 6. d 4 dv v 7. d 9. ( 4 ) 4 ( ) (4 ) d 8. 5 t 4 t dt. 5 d. (u ) du. d. d 4. d Use u-substitution to evaluate in Problems 5-7. 5 4 5. d 6. d 7. ( ) d 8. 4 If f ( ), f (8), and f (7) 5, find f ( ). 9. d d 4 Evaluate ( ) d.. The velocit of a particle moving along a vertical line is given b the equation t v( t). The particle s position at time zero is 4. a. Find an equation for the particle s acceleration at (). b. Find an equation for the particle s position t (). c. At what time(s) is the particle at rest? d. At what time(s) is the particle moving upward? e. For what value(s) of t does the particle s speed equal the particle s velocit? f. Find the total distance traveled b the particle from t = to t = 9. g. Find the interval(s) of time for which the speed of the particle is increasing.. Find equations for the lines tangent to and normal to the graph of 5 when.
98 t. Find the instantaneous rate of change for f ( t) when t. t t. Find the average rate of change for f( t) on [, ]. t 4. Which of the rates of change from Problems and represents: a. the slope of a secant line for the graph of f() t? b. the slope of a tangent line for the graph of f() t? 5. Find the value of c in [, ] such that f() c the average rate of change of t f( t) on [, ]. It is at this t-location that the slopes of what two lines are t the same? (MVT). 6. Differentiate 4 implicitl to find the point(s) where the curve has a. horizontal tangents. b. vertical tangents. 7. (, 7) is a point on the curve of approimate f (.). 8. The graph of f ( ) is shown at right. a. Use the given graph to make f and f number lines. b. Sketch a graph of f which passes through the points (, ) and (, ). f ( ) 5. Use a tangent line to f 9. Use the graph at right to find: a. lim f( ) b. f ( ) c. e. g. lim f( ) lim f( ) lim f( ) 4 d. f () f. f () h. f ( 4). Use the alternate form of the limit definition of the derivative to find f () for f ( ).
99 LESSON 5-6 THE FUNDAMENTAL THEOREM OF CALCULUS, DEFINITE INTEGRALS, CALCULATOR INTEGRATION If f is a continuous function on [a, b], then f ( ) d f ( b ) a a f ( b) f ( a) This relationship is known as the Fundamental Theorem of Calculus. Note: The constant C is not necessar, because b b f ( ) d f ( ) C f a ( b ) C f ( a ) C f ( b ) f ( a ) a Notice the differences between the integration process above, which produces definite integrals, and the previous integration process, which produced indefinite integrals (or antiderivatives). Indefinite Integrals b Definite Integrals f ( ) f ( ) C b f ( ) ( ) ( ) ( ) a a No letters or numbers appear attached a and b (called limits of integration) are to the integral smbol. attached to the integral smbol. a and b b are usuall replaced b numbers in actual problems. Integrating produces an epression Integrating produces a value f ( b) f ( a) f ( ) C which represents a famil which is known as the value of the definite of functions (curves) when written integral. as f ( ) C. ( ) d C ( ) d (a famil of parabolas, if written as C () () () ().5 ) (a number value) The value of a definite integral b f ( ) d ma be thought of as a signed area from a the lower limit a (usuall a left side boundar) to the upper limit b (usuall a right-side boundar), and between the curve of f( ) and the -ais. The value ma be positive, negative, or zero. Calculator Integration: A TI-8 or TI-84 calculator can be used to find the value of a definite integral from a to b b using f ( ) d in the calculate menu or fnint in the math menu. The calculate menu shows a graphical representation of the signed area together with the value of the definite integral.
Eamples: Use the calculate menu to evaluate the following definite integrals.. ( 6 ) d. 6 ( 6 ) 6 d. 5 5 6 d The math menu onl provides the value of the definite integral, but that is usuall all that we need. Most importantl, the math menu gives a more accurate answer. fnint is recommended for all problems from now on. Note: Newer operating sstems have a MATHPRINT setting that simplifies this process. Use the math menu to evaluate: 5 4. 6 d = fnint (abs( 6 ),, 5,5) or if 5 abs( 6 ) is alread entered on our calculator, fnint (,, 5,5) 5. Use the idea of signed area to evaluate d without using a calculator. 6. Set up a definite integral which could be used to find the area of the region bounded b the graph of (shown at right), the -ais, and the vertical lines and. Evaluate without using a calculator. 4 7. d 8. 5 (4 t ) dt 9. 5 d 4. f d (,4) f (4,-4)
START PLUS ACCUMULATION METHOD b Since f ( ) d f ( b ) f ( a ), it follows that f ( b) f ( a) f ( ) d. a a This means a function value can be found as a starting value plus a definite integral. Eamples:. If f and f 4,. If an object s velocit is vt t find f without a calculator. and s 8 find s. b ASSIGNMENT 5-6 The graph of the function f consists of line segments and a semicircle as shown. Evaluate the following using geometr formulas.. f. 4. 4. 4 f d f d For Problems 5 and 6 sketch a graph for each function, and use the idea of signed areas to evaluate these definite integrals using geometr formulas without using a calculator. 5. f ( ) 6. g( ) a. f ( ) d b. f ( ) d a. g ( ) d b. g( ) d Evaluate the definite integrals in Problems 7-5 without using a calculator. 7... ( ) d 8. d 9. ( t ) 4 t dt 4 d. 4 8 d. u u du d * 4. d * 5. d *(Hint: Problems 4 and 5 require u-substitution.)
6. If f( ) is an even function (graph smmetric to the -ais) and a. f ( ) d b. f ( ) d c. f ( ) d d. f ( ) d, find f ( ) d 7. If g ( ) is an odd function (graph smmetric to the origin) and a. g ( ) d b. g ( ) d c. g( ) d d. g( ) d 5, find g( ) d g( ) d Use our calculator to evaluate the definite integrals in Problems 8, 9. Epress answers to or more decimal place accurac. 8. d 9. 6 d. Given f, 4 a. use a calculator to find f d. b. if f, find f ft. If an object s acceleration is at t v, v 5 and 5 find 4. sec 4.. ( t) t t represents the position equation for a particle moving along the -ais. a. Find the velocit equation for the particle. vt () b. Find the acceleration equation for the particle. c. Find the velocit of the particle at t. d. Find the speed of the particle at t. e. At what time(s) is the particle s velocit decreasing? f. Find the displacement of the particle on the interval [, 4]. g. Find the total distance traveled from t to t 4. (Show a velocit number line). You ma wish to review Lesson - for Parts f. and g. 4 h. Find v() t dt without using a calculator. Compare our answer to Part f. i. Use our calculator to find 4 v () t dt. Compare our answer to Part g. *You now have two was to find displacement and total distance. Using definite integrals, displacement b v () t dt and total distance b v() t dt a on the interval a [a, b]. Given a choice of methods, alwas do total distance b evaluating a definite integral on our calculator.
. Find the area between f( ) and the -ais on the interval [, ]. Show an integral set up, and evaluate using a calculator. 4. f ( ), f ( ), and f ( ) a. Without using a calculator, list the domain, an vertical or horizontal asmptotes, the - and -intercepts, and the tpe of smmetr for the graph of f( ). b. Find the -values of the relative etrema of f( ). c. Find the -values of the points of inflection of f( ). d. Sketch f( ) without using a calculator. Check our sketch with a calculator..
4 UNIT 5 SUMMARY Ma/Min Applications: Procedure:. Choose variables and/or draw a labeled figure.. Write a primar equation. Isolate whatever is to be maimized or minimized.. Rewrite with onl one variable on each side. This ma require a secondar equation. 4. Find the domain. 5. Take the derivative, find critical numbers, make a number line, etc. Approimations using a tangent line: Find the equation of a tangent line at a convenient point. Plug in a new -value to find a new -value on the tangent line which is close to a -value on the curve. Antidifferentiation: (Integration) f ( ) d f ( ) C or f ( ) d F( ) C where F( ) f ( ) Indefinite Integrals f ( ) d f ( ) C You might have an initial condition and be able to solve for C. b Definite Integrals f ( ) d f b ( ) f a ( b ) f ( a ) a Start Plus Accumulation f ( b) f ( a) f ( ) d General Power Rule for Integrals (Reverse chain rule for a power function) n n u u ud C, n (Where u is a function of ) n Three Was to Integrate (so far): :. Term b term.. General Power Rule (reverse chain).. u-substitution b Calculator Integration f ( ) d fnint( f ( ),, a, b ) a b a Procedure for u-substitution:. Let u radicand (part inside the smbol).. Solve for.. Differentiate the equation from Step. 4. Find d. 5. Substitute u-epressions for -epressions in the integral. 6. Integrate. 7. Substitute back, so that our final answer is again in terms of. Displacement b v () t dt Total Distance b v() t a a dt Integrals involving absolute value: draw a graph, use geometr.
5 ASSIGNMENT 5-7 REVIEW. Find a and b so that the graph of a b has a relative minimum at (,). 4. The point (,) is on the graph of. Use the equation of a tangent line to approimate the -coordinate when =.9. Use the graph of f shown for problems and 4... Sketch a graph of f 4. Sketch a graph of f which contains the point,. 5. If, find the equation of the tangent line when 6 5 approimate the -coordinate when 5 6. and use it to 6. An interstate driver is traveling 4 miles across a state from south to north without stopping. At noon she notices her speed is 6 miles per hour and her position is at interstate mile marker 4. Note: Interstate mile markers increase from south to north. a. Use this data to write a linear function (local linearization) which could be used to estimate her position as a function of time. Assume t = at noon. b. Approimate her position at : pm. c. Approimate her position at : am. d. What is the domain on which our linear function can be applied?
6 Use the graph of a velocit function for an object moving horizontall shown at the right for problems 7-. 7. Find the object s acceleration at time 5 seconds. 8. Find the speed of the object at time 6 seconds. 9. On which interval of time is the object moving right?. On which interval(s) of time is the object s velocit increasing? vel. in ft sec time in seconds. On which interval(s) of time is the object s speed increasing?. At what time is the object farthest right?. Without using a calculator, find the domain, the intercepts, the vertical asmptote, the end behavior, the relative etrema, and the points of inflection. Then sketch a 4 graph of f. Hint: f and f. 4. Without using a calculator, find the intercepts, the local etrema, and the points of inflection, and then draw a graph of f. 5. Find the point(s) at which the graph of tangents. 6. Find the point(s) at which the graph of tangents. 4 6 has horizontal 4 6 has vertical 7. Find the maimum and minimum points on the graph of 4 6. Determine if the Mean Value Theorem can be applied to f () on the given interval. If it can be applied, find the c-value. If it cannot be applied, eplain wh not. You ma use a calculator. Answer with three or more decimal place accurac. 8. f on,5, on,, 9. f
7. Find the dimensions of the rectangle with maimum area inscribed under the curve as shown. (5,5). A point moves along the curve so that the -coordinate is increasing at the rate of two units per second. At what rate is the -coordinate changing when = 8 units?. Find the absolute etreme values of the function interval,. f on the. If = is a critical number of a function f, and f, does f have a local maimum or a local minimum at =? 4. What is the maimum height (in feet) reached b a ball thrown upward, if the ball s height is given b the position equation st 6t 64t 6? Do not use a calculator. Integrate each of the following. 5. ( ) d 6. t t dt 7. Evaluate the following without a calculator. 8. 4 4 4 d 9. 5 d. 9 5 d d. If f sin and f. 6.5, find.6 f. v t t 4 t. t. The velocit of a moving object is given b. If the position at t = is given b, find. Find the total distance traveled b the object on the interval,4.