Question

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.

Accepted Answer

Step 1: Understand the problem. We need to find the Taylor polynomials of degrees 1, 2, and 3 (denoted as $$p_1$$, $$p_2$$, and $$p_3) $$centered at $$a=1$$ for the function $$f(x) = x^3$. Taylor polynomials approximate a function near a point using derivatives at that point. Step 2: Recall the Taylor polynomial formula centered at a$:

P_n(x) = f(a) + f$$'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n
$$

We will need to compute derivatives of f(x$$)$$ up to the third order and evaluate them at x = 1$. Step 3: Compute the derivatives of f(x$$) = $$x^3$$:

- First derivative: $$f'(x) = 3x^2$
- Second derivative: f$$''(x) = 6x$$
- Third derivative: f$$^{(3)}$$(x) = 6$$

Then evaluate each at $$x=1$$:

- $$f(1) = 1^3$ = 1$$
- $$f'(1) = 3(1)^2$ = 3$$
- $$f''(1) = 6(1) = 6$$
- $$f^{(3)}(1) = 6$ Step 4: Write the Taylor polynomials using the formula:

- Linear polynomial (p_1$):

p_1(x) = f(1) + f$$'(1)(x - 1)
$$

- Quadratic polynomial (p_2$):

p_2(x) = p_1(x$$) + \frac{f''(1)}{2!}(x - 1)^2
$$

- Cubic polynomial (p_3$):

p_3(x) = p_2(x$$) + \frac{f^{(3)}(1)}{3!}(x - 1)^3
$$ Step 5: To approximate values near x=1$, substitute the desired x$ value into p_1$(x)$$ and $$p_2(x$$)$$ for linear and quadratic approximations respectively, and into p_3$(x)$$ for the cubic approximation. This will give increasingly accurate estimates of $$f(x)$$ near $$x=1.$$

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Key Concepts

Taylor Polynomials

Linear and Quadratic Approximations

Derivatives and Their Role in Approximation