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1. A 28 foot ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the ladder is 24 feet from the wall? 2. The radius of a sphere decreases at a rate of 5 m/sec. Find the rate at which the surface area decreases when the radius is 11 m. Answer exactly or round to 2 decimal places. 3. Find the linear approximation L(x) to y=f(x) near x=a for the function. f(x)=sin(x) , a=−32π L(x)= 4. Find the differential and evaluate for the given x and dx. y=sin(3x)x , x=2π , dx=0.25 dy= 5. Find the critical point(s) in the domains of the following function. y=108√x−x2 x= 6. Find the local minima and maxima for the functions over (−∞,∞). Enter DNE if there are no local minimum or maxmimum. Enter coordinates as lists, separated by a comma if there are multiple. y=x2−4x−2 Local minima: Local maxima: 7. Use the Mean Value Theorem to find all points 0<><2 such that f(2)−f(0)=f'(c)(2−0). f(x)=cos(2πx) c= 8. use a calculator to graph the function over the interval [a,b] and graph the secant line from a to b. use the calculator to estimate all values of c as guaranteed by the mean value theorem. then, find the exact value of c, if possible, or write the final equation and use a calculator to estimate to four digits. y=x+1x over [18,16] c= 9. for the following function, f(x)=x4−12x3, determine the following (enter dne if it does not exist): a. intervals where f is increasing or decreasing (in interval notation): increasing: decreasing: b. local minima or maxima of f (enter as a list, separated by commas): local minima at x= local maxima at x= c. intervals where f is concave up and concave down (in interval notation): concave up: concave down: d. the inflection point(s) of f (enter as a list, separated by commas): inflection point(s) at x= 10. evaluate the limit. limx→∞ 4x−1/3x = 11. find two positive numbers, x and y, such that x+y=14 and they minimize x2+y2. x= y= 12. a patient's pulse measures 80 bpm, 100 bpm, then 120 bpm. to determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression (x−80)2+(x−100)2+(x−120)2? what value minimizes it? x= 13. evaluate the limit. limx→a x−a/x2−a2, a≠0 14. evaluate the limit with either hospital’s rule or previously learned methods. limx→0 (4+x)n-4n/x 15. find the antiderivative of the function. f(x)= ex−12x2+sin(x) f(x)= +c 16. evaluate the integral. ∫(sec(x)tan(x)−4x)dx= +c 17. evaluate the integral. ∫4x2+1/x2 dx= +c 18. f(x)=−4(√x)3 f(x)= +c such="" that="" f(2)−f(0)="f'(c)(2−0)." f(x)="cos(2πx)" c="8." use="" a="" calculator="" to="" graph="" the="" function="" over="" the="" interval="" [a,b]="" and="" graph="" the="" secant="" line="" from="" a="" to="" b.="" use="" the="" calculator="" to="" estimate="" all="" values="" of="" c="" as="" guaranteed="" by="" the="" mean="" value="" theorem.="" then,="" find="" the="" exact="" value="" of="" c,="" if="" possible,="" or="" write="" the="" final="" equation="" and="" use="" a="" calculator="" to="" estimate="" to="" four="" digits.="" y="x+1x" ="" over="" [18,16]="" c="9." for="" the="" following="" function,="" f(x)="x4−12x3," determine="" the="" following="" (enter="" dne="" if="" it="" does="" not="" exist):="" a.="" intervals="" where="" f="" is="" increasing="" or="" decreasing="" (in="" interval="" notation):="" increasing:="" ="" decreasing:="" ="" b.="" local="" minima="" or="" maxima="" of="" f="" (enter="" as="" a="" list,="" separated="" by="" commas):="" local="" minima="" at="" x=" " local="" maxima="" at="" x=" " c.="" intervals="" where="" f="" is="" concave="" up="" and="" concave="" down="" (in="" interval="" notation):="" concave="" up:="" ="" concave="" down:="" ="" d.="" the="" inflection="" point(s)="" of="" f="" (enter="" as="" a="" list,="" separated="" by="" commas):="" inflection="" point(s)="" at="" x="10." evaluate="" the="" limit.="" limx→∞="" 4x−1/3x="11." find="" two="" positive="" numbers,="" x="" and="" y,="" such="" that="" x+y="14" and="" they="" minimize="" x2+y2.="" x="y=" 12.="" a="" patient's="" pulse="" measures="" 80="" bpm,="" 100="" bpm,="" then="" 120="" bpm.="" to="" determine="" an="" accurate="" measurement="" of="" pulse,="" the="" doctor="" wants="" to="" know="" what="" value="" minimizes="" the="" expression="" (x−80)2+(x−100)2+(x−120)2?="" what="" value="" minimizes="" it?="" x="13." evaluate="" the="" limit.="" limx→a="" x−a/x2−a2,="" a≠0="" 14.="" evaluate="" the="" limit="" with="" either="" hospital’s="" rule="" or="" previously="" learned="" methods.="" limx→0="" (4+x)n-4n/x="" 15.="" find="" the="" antiderivative="" of="" the="" function.="" f(x)="ex−12x2+sin(x)" f(x)="+C" 16.="" evaluate="" the="" integral.="" ∫(sec(x)tan(x)−4x)dx="+C" 17.="" evaluate="" the="" integral.="" ∫4x2+1/x2="" dx="+C" 18.="" f(x)="−4(√x)3" f(x)="">