# Math Insight

### The tangent line as a linear approximation

Math 201, Spring 19
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Total points: 3
1. Let $f(x)=x^{2}$. We will calculate linear approximations of $f$ around different points. But first, sketch the graph of $f$ below.
Feedback from applet
function:
linear approximation locations:
linear approximation slopes:
number of linear approximations:
1. As a first step toward calculating linear approximations, calculate the derivative: $f'(x)$ =

The linear approximation (or tangent line) to $f$ around $x=2$ is a line through the point $(2,f(2))$ with slope $f'(2)$: $y=f'(2)(x-2) + f(2)$. Write the equation for this linear approximation.
$y=$

Sketch this linear approximation on the above graph.

2. 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 $L_{ 2 }$.)

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

$f(2)=$
, $L_{ 2 }(2)=$

$f(1.9)=$
, $L_{ 2 }(1.9)=$

$f(2.1)=$
, $L_{ 2 }(2.1)=$

$f(10)=$
, $L_{ 2 }(10)=$

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

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

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 $f$.)

3. 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: $x=-2, -1, -1.1, -0.9, 10$.
$f(-2)=$
, $L_{ -1 }(-2)=$

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

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

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

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

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

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

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 $f$.)

2. We can find linear approximations of functions of different variables than $x$. For example, let $g(t)=t^n e^{1-t}$, where $n$ is some number.
1. Let $n=1$, so $g(t)=$
and $g'(t)=$

Find the equation for the tangent line of $g$ at $t=0$.
$y=$

Find the equation for the linear approximation of $g$ around $t=1$
$L_{1}(t)=$

On the below graph of $g$, plot the tangent lines corresponding to these two linear approximations.

Feedback from applet
number of tangents:
parameter n:
tangent locations:
2. Let $n=2$, so $g(t)=$
and $g'(t)=$

Find the equation for the tangent line of $g$ at $t=0$.
$y=$

Find the equation for the linear approximation of $g$ around $t=1$
$L_{1}(t)=$

Find the equation for the linear approximation of $g$ around $t=2$
$L_{2}(t)=$

On the below graph of $g$, plot the tangent lines corresponding to these three linear approximations.

Feedback from applet
number of tangents:
parameter n:
tangent locations:

3. Let $h(x)=\ln{\left (x^{2} + 1 \right )}$. Find linear approximations for $h(x)$ around $x=-1$, $x=0$, and $x=1$.

Around $x=-1$: $L_{-1}(x)=$

Around $x=0$: $L_0(x)=$

Around $x=1$: $L_1(x)=$