# Homework-6_solution - Math1013 Calculus I, Fall 2012...

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Math1013 Calculus I, Fall 2012Name:Tsz Hing LOHomework-6 : Due 12/07/2012 at 11:50pm HKTThis homework set covers the basics of antiderivatives (indefinite integrals)and initial value problems, Riemann sums and definite integrals, FundamentalTheorem of Calculus, and the Substitution Rule.1. AntiderivativesZf(x)dx=F(x)+C←→dFdx=f2. Riemann sums are just sum of +ve/-ve rectangular area over subinter-vals, which lead to definite integrals by limit considerations.3. Fundamental Theorem of Calculus:Zbaf(x)dx=F(b)-F(a)if4.(4 pts) A particle is moving with accelerationa(t) =24t+16. its position at timet=0 iss(0) =6 and its veloc-ity at timet=0 isv(0) =1. Hint: this is the same problemas the first. Treat acceleration as the second derivative and ve-locity as the first derivative, with the distance being the originalfunction.
4. Substitution rule: turning a complicated integralZf(x)dxinto aneasier oneZg(u)duby an appropriate choice of substitution:u=g(x),du=g0(x)dx.Give 4 or 5 significant digits for numerical answers.For most problemswhen entering numerical answers, you can if you wish enter elementary expres-sions such as 32 or 3**2 instead of 9, sin(3*pi/2)instead of -1,e(ln(3))instead of 3,(1+tan(3))*(4-sin(5))6-15/8 instead of 12748.8657, etc.1.(4 pts)Find the derivative off(x) =-x2+4949x+Cto complete thefollowing integration formula:
2.(6 pts) Calculate the following antiderivatives:(a)Z8t-5t5+2dt=+C.(b)Z1u1/4+2udu=+C.(c)Z16x6dx=+C.Correct Answers:3.(6 pts) Calculate the following antiderivatives:(a)Z10dx=+C.
xZ
(c)5exdx=+C.

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