137_PS6_Taylor.pdf - Problem Sets u2013 Taylor series and...

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Problem Sets – Taylor series and approximationThese problem sets contain additional practice problems that correspond to the lecture material. There are hundredsmore in the calculus textbooks on reserve at the library, or on the Davis shelves close to QA303.Maclaurin and Taylor series1. Determine the first three non-zero terms in the Maclaurin series for the following.a)ecosx=e×ecosx-1b) ln (cosx) = ln (1 + [cosx-1])c) coshx=ex+e-x2Determine the first three non-zero terms of the Taylor series for the following functions about the given points.d)f(x) = sinx,x0=π2,e)f(x) =ex,x0= 3Estimation2. Estimate the value ofearcsinx+ ln1-xeatx= 3×10-4Limits3. Evaluate the following limits using Maclaurin series.a) limx01x-1ex-1b) limx0x2-sin2xx2sin2xc) limx0sin2(2x)x2Numbers and polynomials4. In combinatorial mathematics, theBell numbersbncount the possible partitions of a set. For example,b3= 5because the 3-element set{a, b, c}can be partitioned in 5 distinct ways:{{a},{b},{c}},{{a, b},{c}},{{a, c},{b}},{{

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