Find the quadratic polynomial g(x)-ax^{2} + bx + c text{which best fits the function} f(x)=e^{x} text{at} x=0, text{in the sense that} g(0)=f(0), text{and} g'(0)=f'(0), text{and} g''(0)=f''(0). Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?

aflacatn

aflacatn

Answered question

2021-01-30

Find the quadratic polynomial g(x)ax2 + bx + c which best fits the function f(x)=ex at x=0, in the sense that g(0)=f(0), and g(0)=f(0), and g(0)=f(0). Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?

Answer & Explanation

delilnaT

delilnaT

Skilled2021-01-31Added 94 answers

Step 1 The functions f(x) and g(x) are: f(x)=ex
g(x)=ax2 + bx + c
f(x)= d(f(x))dx= d(ex)dx

Using formula for derivative of exponential function: f(x)=ex
f(x)= d(f,(x))dx= d(ex)dx

Using formula for derivative of exponential function: f(x)=ex
g(x)= d(g(x))dx= d(ax2 + bx + c)dx

Using Theorem 3.2: g(x)= d(ax2)dx + d(bx)dx + d(c)dx

Using Theorem 3.1: g(x)=a × d(x2)dx + b × d(x)dx + 0

Step 2 Using Power Law: g(x)=a × 2 × x2  1 + b × 1 + 0
g(x)=2ax + b
g(x)= d(g(x))dx= d(2ax + b)dx

Using Theorem 3.2: g(x)= d(2ax)dx + d(b)dx

Using Theorem 3.1:

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