MAC 2311C Lecture 28: MAC 2311, Lecture 28 (5.4 Indefinite Integrals) Part 1

1. Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible. A particle is moving with af)-sint +3cost, find the position, x()of the particle given x(0)-0, v(0)-2, where v(0) is the initial velocity and t is time in seconds. 2. 3. Use Newton's Method to find 2 correct to five decimal places 4. Find the definite Integrals b) tCosSee Sires) dat 5. Find the Indefinite Integrals a) fr -2sinx+ Inr de b) sina) d sin(In x d) tanx Incosx) d 6. Find the area under the curve, f(x)= sin x at the interval | π, π | with n = 5 using a) Right-Sum Method b) Left-Sum Method c) In which Method, you have overestimated the actual value, why? 7. Find the derivative of the function: 3x+1 b) Extra Credit: y=l., sin(t*)dt fr(s)-dr , where g (x)-厂1+sin(?) dt , find f'(r/2). i. If f(x)- ii. True or false, explain 1+1 a) If f(x)-g(x) for 0

(z)236(z + 3). (d) λ (z) = δ(x + 1) + 36(x-1) (d) Ala)=6(z+ 1) +36 (c) λ(z) = 56(z)-W(z + 3). 5.2. Evaluate the following integrals: Use e* sin (c) / e*sin (x3 + 1)dr. (e)/ sin-1(1/r)o(2,4-1)dr. (d) / (f) / sin ( )6(x4 + 1) dz. cos(nz)#6a 2-x-1) dz, and J-0.1 an ㄫㄨ ln(1 + x) sin whe 5.6 5. 5.3. Show that

MATH 31S In order to compute the above quantities, one needs to be able to evaluate infinite inte- grals. We use the following approach. For any real number a define f(x)dx= linn | f(z)dz. when the limit exists and is finite, then the integral is said to be convergent, otherwise it is divergent. Similarly f(x)dェ=tlinnol, f(x)dx. (2) Problem 3 Determine whether the following integrals are convergent or divergent. If -00 convergent, determine their value: dr,dr cos(x)dr. (3) Problem 4 Dctcrminc for which p >0, the integral is convergent and divergent.