= f (c) f (c) the height of the rectangle guaranteed by the MVT for integrals.
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1 Get Rey: Given (t) = 8t n v() = 6, fin the isplcement n istnce of the oject from t= to t= If () = 4, fin the position of the prticle t t= I. Averge Vlue of Function Wht oes represent? Cn we rw rectngle with se= which is ectly equl to this integrl? Men Vlue Theorem for Integrls If f is continuous on the close intervl [,], then there eists numer c on [,] such tht = f (c)( ). (The re uner the curve etween n cn e foun y multiplying the se of the rectngle (- ) times the height of the rectngle t some point (c, f (c)) ). = f (c) Therefore, ( ) n ( ) is the verge vlue of the function n f (c) is the height of the height of the rectngle gurntee y the MVT for integrls.
2 . Given f () = + on the intervl [,], fin:. the verge vlue of the function. the vlue of c gurntee y the MVT for integrls. Given f () = sin on the intervl [,π], fin:. the verge vlue of the function. the vlue of c gurntee y the MVT for integrls II. The Secon Funmentl Theorem of Clculus Use the FTC to evlute (t 4t +)t Hint: F() F() Tke the erivtive of your nswer. Oservtions: III. The Secon FTC If f is continuous on n open intervl I contining, then, for every in the intervl, u [ f (t)t] = f () n [ u f (u)t] = f () F() = f (t)t à If, then F '() = f () The erivtive of n integrl is the originl integrn (ut with the vrile chnge).
3 IV. Emples of the Secon FTC: 3. [ t ]t. [ t sint ]t [ cost ]t π 3. tn(t 3 sin ) t t 4 t cost t V. Fin the verge vlue of the function f on the given intervl. NO CALCULATORS! f () = cos on [, π. f () = on [,3]. ] 3. f () = on [,9] f () = on [,5] VI. Fin the vlue of gurntee y the MVT for Integrls. No clcultors on # n #!. f () = 4 on [,]. f () = 4 on [,3] 3. f () = 3 + on [,] f () = sin( ) on [, π ]
4 VII. Mie. In city, the temperture ( F ) t hours fter 9: AM is pproimte y the function T (t) = 5 +4sin( πt ). Fin the verge temperture uring the time perio 9AM to 9PM. F() =. Show the Secon FTC hols for t t. Tke the integrl, then tke the erivtive. 3. Use the Secon FTC to fin the erivtives of the following functions. f () = (t +) t g() = t 3 +t f () =.. c. + t t g() = cos(t )t f () = (t 3 )t 4 π. 4 e. y = (t 3 + t +)t Fin the intervl on which the curve is concve up. Justify your nswer.
5 5. Time(min) F (Temp egrees Fhrenheit) The temperture of pot of soup t time t is moele y strictly incresing, twice- ifferentile function F, where F(t) is mesure in egrees Fhrenheit n t is mesure in minutes. At time t =, the temperture of the soup is 4 F. The soup is hete for 5 minutes, eginning t time t =. Vlues of F(t) t selecte times t for the first 5 minutes re given in the tle ove. 5. Use the t in the tle to evlute F'(t)t. Using correct units, interpret its vlue n eplin its mening.. For t 3, eplin the mening of 5 5 F(t)t in the contet of this prolem. 5 c. Estimte the vlue of 5 F(t)t using right Riemnn sum with five suintervls. Is this n over or uner estimte? Eplin your resoning. 5. Estimte the vlue of 5 F(t)t using left Riemnn sum with five suintervls. Is this n over or uner estimte?? Eplin your resoning. e. Estimte the vlue of 5 5 F(t)t using trpezoil Riemnn sum with five suintervls.
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