This linear approximation was calculated from the value and slope of $f$ at $x=2$, but we can use it to estimate values of $f$ for different values of $x$. To emphasize this fact, let's denoted the linear approximation by $L_{ 2 }(x)$. Rewrite the linear approximation using this notation.

$L_{ 2 }(x) =$

Let's compare the values of the actual function $f(x)$ and this particular linear approximation $L_{ 2 }(x)$ for the following values: $x=1, 2, 1.9, 2.1, 10$. (For all these values of $x$, we are still using the linear approximation calculated from $x=2$, i.e., equation you just calculated above. That's why we denote it $L_{ 2 }$.)

$f(1)=$

, $L_{ 2 }(1)=$

$f(2)=$

, $L_{ 2 }(2)=$

$f(1.9)=$

, $L_{ 2 }(1.9)=$

$f(2.1)=$

, $L_{ 2 }(2.1)=$

$f(10)=$

, $L_{ 2 }(10)=$

(Calculate your answers to at least 4 significant digits.)

For which of these values is $L_{ 2 }(x)$ close to $f(x)$ (within a few percent of $f(x)$)?

(Separate multiple answers by commas.)

For which of these values is $L_{ 2 }(x)$ a bad approximation to $f(x)$ (deviating more than 50% from $f(x)$)?

(Separate multiple answers by commas.)

The linear approximation $L_{ 2 }(x)$ calculated from $x=2$ is a good approximation to $f(x)$ when $x$ is

. (This should correspond to values of $x$ where the tangent line calculated at $x=2$ is close to the graph of $f$.)

##### Hint

Your answers for $L_{ 2 }(x)$ should be exactly the same as for the previous part. We can write a linear approximation as $y=\ldots$ to emphasize it is an equation for the tangent line or as $L_{ 2 }(x)=\ldots$ to emphasize we can use the linear approximation to estimate values of $f(x)$ for different values of $x$.

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We can also calculate linear approximations around other values of $x$. Calculate the linear approximation around $x=-1$. Write it as the equation of a tangent line:

$y=$

Also, write it using $L_{ -1 }(x)$. The $-1$ in $L_{ -1 }(x)$ means we calculate the linear approximation at $x=-1$.

$L_{ -1 }(x)=$

Sketch the graph of this second linear approximation on the above graph.

Compare the value of $f(x)$ to the value of the linear approximation $L_{ -1 }(x)$ (calculated at $x=-1$) for the following values: $x=-2, -1, -1.1, -0.9, 10$.

$f(-2)=$

, $L_{ -1 }(-2)=$

$f(-1)=$

, $L_{ -1 }(-1)=$

$f(-1.1)=$

, $L_{ -1 }(-1.1)=$

$f(-0.9)=$

, $L_{ -1 }(-0.9)=$

$f(10)=$

, $L_{ -1 }(10)=$

(Calculate your answers to at least 4 significant digits.)

For which of these values is $L_{ -1 }(x)$ close to $f(x)$ (within a few percent of $f(x)$)?

(Separate multiple answers by commas.)

For which of these values is $L_{ -1 }(x)$ a bad approximation to $f(x)$ (deviating more than 50% from $f(x)$)?

(Separate multiple answers by commas.)

The linear approximation $L_{ -1 }(x)$ calculated from $x=-1$ is a good approximation to $f(x)$ when $x$ is

. (This should correspond to values of $x$ where the tangent line calculated at $x=-1$ is close to the graph of $f$.)

##### Hint

To sketch the linear approximation online, change $n_t$ (for number of tangent lines) to 2, then drag the red points so that the red line is the linear approximation to $f$ at $x=-1$.

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