Home > APCALC > Chapter 10 > Lesson 10.3.2 > Problem10-143

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 $e^{0.7}$.

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

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

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

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

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

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

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