Question

2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b) Find the second order Taylor expansion of f at 1, namely f(x) = ao + ai (z-1) + a2(z-1)2 + R2(x where R2 is the remainder. You should find ao, a1, a2, and (c) Use the Taylor expansion found above to estimate the valu that the error in this estimation (i.e., R2(0.9)I) is at most 3. Evaluate the following integrals. (b) sin1120 sin3 2dz. (c) sin Vada. (d)广(lnxy 부dz. 4. Define In = / xnV1_z?dx for integer n 20 (a) Compute 1,-I v1-32da and 11 = | xyT_z?dz. (b) Using integration by parts, express In+2 in terms of In (c) Compute I4 and Is.

Answer 1

2. 久 IIT

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