Math Insight

The tangent line as a linear approximation

Math 1241, Fall 2018
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Due date: Oct. 19, 2018, 11:59 p.m.
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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 $L_{ 2 }$.)

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

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

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

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

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

      (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 $f$.)

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

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

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

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

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

      (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 $f$.)

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