§1.8 The tangent line approximation (continued)

The local linearization

Suppose that a function $y = f(x)$ has its tangent line approximation given by $L(x) = 3 - 2(x-1)$ at the point $(1,3)$, but we do not know anything else about the function $f$. How can we estimate the value of $f(x)$ for $x$ near $1$?

Compute the values of $L(-1)$ and $L'(-1)$.

What must be the values of $g(-1)$ and $g'(-1)$? Why?

Do you expect the value of $g(-1.03)$ to be greater than or less than the value of $g(-1)$? Why?

Use the local linearization to estimate the value of $g(-1.03)$.

Suppose that you also know that $g''(-1) = 2$. What does this tell you about the graph of $y = g(x)$ at $a = -1$?

For $x$ near $-1$, sketch the graph of the local linearization $y = L(x)$ as well as a possible graph of $y = g(x)$ on the axes provided.

Find a formula for the tangent line approximation, $L(x)$, to $f$ at the point $(2,-1)$.

Use the tangent line approximation to estimate the value of $f(2.07)$. Show your work carefully and clearly.

Sketch a graph of $y = f''(x)$ on the righthand grid in the provided figure; label it appropriately. Write a sentence to explain why your graph looks the way it does.

Is the slope of the tangent line to $y = f(x)$ increasing, decreasing, or neither when $x = 2$? Explain.

Sketch a possible graph of $y = f(x)$ near $x = 2$ on the lefthand grid in the provided figure. Include a sketch of $y=L(x)$ (found in part (a)). Explain how you know the graph of $y = f(x)$ looks like you have drawn it.

Does your estimate in (b) over- or under-estimate the true value of $f(2.07)$? Why?

Compute $f(3)$, $f'(3)$, $g(3)$, and $g'(3)$. What do you observe?

Determine $L(x)$, the local linear approximation to $f(x)$ at $x = 3$. Explain why this function is also the local linear approximation to $g(x)$ at $x = 3$.

Complete the following table of values for $f(x)$, $g(x)$, and $L(x)$:

$x$ | $2.9$ | $2.99$ | $2.999$ | $3$ | $3.001$ | $3.01$ | $3.1$

$f(x)$ | | | | | | |

$g(x)$ | | | | | | |

$L(x)$ | | | | | | |

What do you notice about how $L(x)$ approximates $f(x)$ and $g(x)$ near $x = 3$?