CPM Homework Help : APCALC Problem 10-143

Write the equation of a quadratic function $p(x)=ax^2+bx+c$ which best approximates $f(x) = e^x$ about the point $(0, 1)$ . Then calculate the error if the quadratic function is used to approximate $e^{0.7}$ .

Step 1:

$f^\prime(x)=e^x\text{ }\Rightarrow\text{ }f^\prime(0)=1$

$f^{\prime\prime}(x)=e^x\text{ }\Rightarrow\text{ }f^{\prime\prime}(0)=1$

Step 2:

$p(0)=a(0)^2+b(0)+c=1\text{ }\Rightarrow\text{ }c=1$

$p^{\prime\prime}(0)=2a=f^{\prime\prime}(0)=1\text{ }\Rightarrow\text{ }a=0.5$

$p^\prime(0)=2a(0)+b=f^\prime(0)=1\text{ }\Rightarrow\text{ }b=1$

Step 3:

$p(x)=0.5x^2+x+1$

Step 4:

Error: $p(0.7) - f(0.7)$