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Math1013 Calculus I, Fall 2012Name:Tian TIANHomework-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.2. 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:4. Substitution rule: turning a complicated integralZg(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 3∧2 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+99x+Cto complete thefollowing integration formula:Rdx=-√x2+9Answer(s) submitted:•(incorrect)2.(6 pts) Calculate the following antiderivatives:(a)Z8t-5t3+9dt=+C.(b)Z1u5/4+5.5√udu=+C.(c)Z15x3dx=+C.Answer(s) submitted:•3.(6 pts) Calculate the following antiderivatives:Answer(s) submitted:•••(incorrect)4.(4 pts) A particle is moving with accelerationa(t) =12t+18. its position at timet=0 iss(0) =11 and its velocityat timet=0 isv(0) =2. Hint: this is the same problem as thefirst. Treat acceleration as the second derivative and velocity asthe first derivative, with the distance being the original function.What is its position at timet=5?