Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or. This is not complete list of the mteril tht you should know for the course, but it is good indiction of wht will be emphsized on the exm. A thorough understnding of ll of the following concepts will help you perform well on the exm. Some plces to find problems on these topics re the following: in the book, in the slides, in the homework, on quizzes, nd WebAssign. Optimiztion: Section.7 Optimiztion is pplied clculus. For this course, optimiztion problems re solved using the Closed Intervl Method, the st Derivtive Test, or the nd Derivtive Test. Solving n Optimiztion Problem: () Wht re you ttempting to optimize? Wht conditions limit this process? () Digrm (if pplicble) nd fix nottion. (3) Wht re the constnts nd vribles? Wht is known nd unknown? () Find the function which is to be optimized. (5) Express the limiting conditions s equtions. (6) Optimize!. Design cylindricl cn of volume 9cm 3 so tht it uses the lest mount of metl.. A Normn window hs the shpe of rectngle surmounted by semicircle. If the perimeter of the window is 3 ft, find the dimensions of the window so tht the gretest possible mount of light is dmitted. 3. Find positive number such tht the sum of the number nd its reciprocl is s smll s possible.. Find the mximum length of pole tht cn be crried horizontlly round corner joining corridors of widths ft nd 3 ft. 5. Find the eqution of the line through P = (,) such tht the tringle bounded by this line nd the xes in the first qudrnt hs miniml re. Newton s Method: Section.8 To pproximte root of f (x), choose n initil vlue x. Generte successive pproximtions of the root through the eqution x n+ = x n f (x n) f (x n ) Sometimes, Newton s Method does not pproximte root. (I) Choices of x which re criticl points do not pproximte roots s Newton s Method fils. (II) Certin choices of x leds towrds n symptote.. Use four itertions of Newton s Method to pproximte 3.. Use four itertions of Newton s Method to pproximte 3.
3. Find the roots of f (x) = x 3 5x+ using Newton s Method. Use Clculus to sketch the grph to id in mking good initil choices. Integrl Bsics: Section 5. nd 5. The net re A of the region S tht lies under the grph of the continuous function f is pproximted by n rectngles: Divide the domin into segments of equl length, x = b n. Inside ech segment choose vlue x i. Form rectngle x f (x i ) on ech segment. Then A is pproximted by A ( f (x ) x + f (x ) x +... + f (x n ) x) The definite integrl of f on the intervl [,b] is = lim n n i= f (x i ) x where x = b n nd x i = + i x, provided tht this limit exists. (i) If f is continuous on [,b], or if f hs only finite number of jump discontinuities, then f is integrble on [,b]. (ii) The definite integrl clcultes net re. To find the totl re contined between function nd the x-xis, clculte f (x) dx. (iii) For constnt c, cdx = c(b ). (iv) = c + c
(v) (vi) (vii) (viii) = b =. ( f (x) ± g(x))dx = c = c ± g(x) dx (ix) If f (x) g(x), then g(x) dx. (x) If m f (x) M on the intervl [,b], then m(b ) M(b ). Exercises:. Prove tht 6 3 x dx.. Express 9 5 s single integrl. 3. Evlute the following integrls ssuming tht () = = (c) (d). Stte whether true or flse. If flse, sketch the grph of counterexmple. () If f (x) >, then If >. >, then f (x) >. = 7 Antiderivtives nd the Fundmentl Theorem of Clculus: Section 5.3-5.6 For the function f nd vlue, the cumultive re function A f (x) is the net re under the curve f on the intervl [,x]. A f (x) = 3 f (t)dt
Let f (x) be continuous on [,b] nd let F be n ntiderivtive of f. Let A f (x) = f (t)dt. (FTC Prt I) (Evlution) d ( A f (x) ) = f (x). dx = F F(). The Fundmentl Theorem of Clculus Prt I shows tht every continuous function hs n ntiderivtive - nmely, its re function (with ny lower limit). To differentite the function F(x) = Fundmentl Theorem of Clculus nd the Chin Rule: F (x) = f (g(x))g (x). g(x) f (t)dt, use the. Show tht F(x) = tn (x) nd G(x) = sec (x) hve the sme derivtive. Wht cn you conclude bout the reltion between F nd G?. A 9kg rocket is relesed from spce sttion. As it burns fuel, the rocket s mss decreses nd its velocity increses. Let v(m) be the velocity s function of mss m. Find the velocity when m = 79kg if dm dv = 5m. Assume tht v(9) = m s. 3. A hmmer is dropped nd it flls for seconds before hitting the ground. Determine how fr it flls, ssuming grvity is the only force cting upon the hmmer.. Clculte the following derivtives: d x () (t 5 9t )dx dx d cos(x) t dx 6 t + dt d x (c) t dt dx x 5. The following is grph of y = f (x). Let A(x) = f (t)dt nd B(x) = f (t)dt. () Find the min nd mx of A on [,6]. Find the min nd mx of B on [,6]. (c) Find formuls for A(x) nd B(x) vlid on [3,]. (d) Find formuls for A(x) nd B(x) vlid on [,6]. 6. Sketch the grph of n incresing function f such tht both f (x) nd 7. Clculte 3 f (x) dx where f (x) = { x x x 3 x > f (t) dt re decresing.
8. Clculte the following definite integrls: () (c) 9 5 x dx (3x 9e 3x )dx x x + 3 dx Integrtion with Substitution: Section 5.7 The Substitution Rule is the reversl of the Chin Rule. Use substitution when the integrnd hs the form f (g(x))g (x). If F is n ntiderivtive of f, then f (g(x))g (x)dx = F(g(x)) +C When substituting u = g(x), the differentil of g(x) is relted to dx by du = g (x)dx. The Substitution Method is expressed by the Chnge of Vribles Formul: f (g(x))g (x)dx = f (u)du The Chnge of Vribles for Definite Integrls: f (g(x))g (x)dx = g g() f (u)du. Evlute the following definite nd indefinite integrls: () (x 5) 9 dx (f) sin 8 (x)cos(x)dx dt (g) sec (x)e tn(x) dx t 7 (c) x x 3 + dx (h) (x 9) 3 dx e (d) x 5 ln(x) x 3 + dx (i) dx x π (e) cot(x) dx (j) cos 3 (x)sin(x)dx. Use substitution to evlute the integrl in terms of f (x): f (x) () f (x) dx x f ( x + )dx 5