Review Taylor polynomials by constructing local approximations from derivatives and evaluating them near the centre, with terms kept to the requested degree.
Let y=f(x) be the particular solution to the differential equation dxdy=(3−x)y2 with initial
condition f(1)=-1.
Write the second-degree Taylor polynomial for f about x=1.
B Write the second-degree Taylor polynomial for f about x=1.
| f′(1)=2 and f′′(1)=−9 | Two terms | Point 4 (P4) |
|---|---|---|
| P2(x)=−1+2(x−1)−29(x−1)2 | Remaining term | Point 5 (P5) |
Scoring Notes for Part B
- P4 and P5 can be earned with an answer consistent with incorrect values of f′(1) and f′′(1)
imported from part A.
- Any terms of degree greater than two or " +… " does not earn P5.
- A response of −1+2(x−1)−29(x−1)2 earns P4 and P5, regardless of any subsequent algebraic
simplification.
- A response that does not present the polynomial as powers of (x-1) but instead presents a correct
expanded/simplified form of the polynomial (e.g., −29x2+11x−215 ) earns P4 but not P5.
Let y=f(x) be the particular solution to the differential equation dxdy=y⋅(xlnx) with initial condition f(1)=4. It can be shown that f′′(1)=4.
Write the second-degree Taylor polynomial for f about x=1. Use the Taylor polynomial to approximate f(2).
Write the second-degree Taylor polynomial for f about x=1. Use the Taylor polynomial to
approximate f(2).
f′(1)=dxdy(x,y)=(1,4)=4⋅(1ln1)=0
The second-degree Taylor polynomial for f about x=1 is
f(1)+f′(1)(x−1)+2!f′′(1)(x−1)2=4+0(x−1)+24(x−1)2.
Polynomial
1 point
f(2)≈4+2(2−1)2=6
Approximation
1 point
Scoring notes:
- The first point is earned for 4+1!4⋅ln1(x−1)1+2!4(x−1)2 or any correctly simplified equivalent expression. A term involving (x-1) is not necessary. The polynomial must be written about (centered at) x=1.
- If the first point is earned, the second point is earned for just " 6 " with no additional supporting work.
- If the polynomial is never explicitly written, the first point is not earned. In this case, to earn the second point supporting work of at least " 4+2(1) " is required.
Total for part (a) 2 points