Section 4.11 : Linear Approximations
4. Find the linear approximation to f(t)=cos(2t) at t=12. Use the linear approximation to approximate the value of cos(2) and cos(18). Compare the approximated values to the exact values.
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Start SolutionWe’ll need the derivative first as well as a couple of function evaluations.
f′(t)=−2sin(2t)f(12)=cos(1)f′(12)=−2sin(1) Show Step 2Here is the linear approximation.
\require{bbox} \bbox[2pt,border:1px solid black]{{L\left( t \right) = \cos \left( 1 \right) - 2\sin \left( 1 \right)\left( {t - \frac{1}{2}} \right) = 0.5403 - 1.6829\left( {t - \frac{1}{2}} \right)}}Make sure your calculator is set in radians! Remember that we use radians by default in this class.
Show Step 3Now, if we want to approximate \cos \left( 2 \right), that is equivalent to evaluating f\left( 1 \right) = \cos \left( 2 \right), we need to evaluate the linear approximation at t = 1. Likewise, to approximate \cos \left( {18} \right) we need to evaluate the linear approximation at t = 9.
So, here are the approximations of the values along with the exact values.
\begin{align*}L\left( 1 \right) & = - 0.301169 & \hspace{0.5in}f\left( 1 \right) & = - 0.416147 & \hspace{0.5in}{\mbox{% Error : }} & 27.6292\\ L\left( 9 \right) & = - 13.7647 & \hspace{0.5in}f\left( 9 \right) & = 0.660317 & \hspace{0.5in}{\mbox{% Error : }} & 2184.56\end{align*}So, as we might have expected the farther from t = \frac{1}{2} we got the worse the approximation is. Recall that the approximation will generally be more accurate the closer to the point of the linear approximation.