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Use linear approximation to estimate $f(5.1)$ given that $f(5)=10$ and $f^{\prime}(5)=-2$
$\quad L(x)=-2 x+20$ and $f(5.1)=L(5.1)=9.8$
Calculus 1 / AB
Chapter 4
Applications of the Derivative
Section 6
Linear Approximation and Differentials
Derivatives
Differentiation
Campbell University
Harvey Mudd College
University of Michigan - Ann Arbor
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in this problem, we need to take our linear approximation formula substituted known values to get our linear approximation equation for this set up and then a substitute in 5.1 into this equation to get our answer of fo 5.1. So let's start by substituting in our known values in this case, A is going to equal five because that is the given value. We haven't that is given value for this problem. So our Ella Vex is going to be F five plus of prime of five times ex my size now each effort five and a terrified of we know what those are 10 and negative two, respectively. So we can rewrite this as l've X equals 10 minus two times X minus five, I wrote minus students that have plus negative, too, because the negative just carries over the positive. So it's really just 10 minus two times X minus five and simplifying this down. We get our final at l A X equation, which is negative two X plus 20. So this is our final of X, and what we can now do have solved for F of 5.1 is Since this is the linear approximation at f of five, we can substitute 5.1 and for X and get our final answer. So L a five point warned equals F of 5.1, which equals negative two times 5.1, uh, plus 20. And this equals 9.8, which is our final answer.
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