# 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?

Find the quadratic polynomial Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?
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Step 1 The functions f(x) and g(x) are: $f\left(x\right)={e}^{x}$

Using formula for derivative of exponential function: ${f}^{\prime }\left(x\right)={e}^{x}$

Using formula for derivative of exponential function: $f{}^{″}\left(x\right)={e}^{x}$

Using Theorem 3.2:

Using Theorem 3.1:

Step 2 Using Power Law:

Using Theorem 3.2:

Using Theorem 3.1: