Calculus 1 Lab 2
Linear Approximation
This lab will cover ‘‘linear approximations’’: what are they, how do you compute
them, and how are they useful.
Unless stated otherwise, compute all values to decimal places in this lab.
The goal of this section of the lab is to explore and understand linear approximations and what they should be. Linear approximations will be important for topics in later courses such as ‘Newton’s Method’ (Calculus II), and are the reason why in physics, sometimes you say .
Disclaimer on this lab:
In this lab you will be approximating function values using a technique called linear
approximation. Most of the examples discussed in this lab use familiar functions.
Although we may be able to evaluate the function directly and there is no
need to approximate, we will use these examples to learn how to find linear
approximations.
In biology systems, the function you are trying to evaluate is often not known; you
only know the way the system changes (i.e. the derivative of the desired function)
and one initial condition (i.e. the output of the desired function at one x-value). If
you want to find the function at a different value than the one given, you may need
to approximate the function value.
The main idea of solving such a problem is to create a linear approximation of the
function at a point you know. From there, you can use the linear approximation to
estimate function values for points nearby. Linear approximation is a vital tool, so
we will build our intuition for linear approximations on familiar functions.
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Now …to the material.
Suppose you want to approximate the function at .
Let’s explore what properties we would want from this approximation.
Objective 1: Play with the (slope) and (y-intercept) sliders to find a ’good’ linear
approximation of at . This is not a question that can be correct or incorrect; it is
meant to build your intuition on linear approximations.
Hint: In the desmos window below, go to an empty box in the desmos grapher and type ‘’. This will make a ‘dot’ at the point on that we want to approximate at. You may also want center the point and zoom in to get an even better approximation.
As approximations of the curve at ,
- the error of Line A (given by , in orange) is
- the error of Line B (given by , in red) is
- the error of Line C (given by , in blue) is
- the error of Line D (given by , in green) is
This establishes an important property of a linear approximation of a function at a
given point: the best linear approximation should agree with the point at which you
are approximating. In other words, if you are approximating a function
at with a line , then the line should be such that ; the output of the line
is the same as the function’s output at the -value you are approximating
at.
Question: What is the difference between two lines that both have no error where you are approximating at? Consider the graph of (in black), and three other linear approximations.
Given the new graph above, then as approximations of the curve at ,
- the error of Line B (given by , in red) is
- the error of Line C (given by , in blue) is
- the error of Line G (given by , in green) is
Hint: Though you can do the computations by hand, use the desmos calculator
below to help with the computations. Note that that is given for you. Alter this to
get what you’re looking for!
Click the ‘Reveal Hint’ button to learn how to evaluate the functions using the
desmos calculator.
Use the desmos calculator above to answer the following questions.
As approximations of the curve at ,
- the error of Line B (given by , in red) is
- the error of Line C (given by , in blue) is
- the error of Line G (given by , in green) is
Warning: ‘Near’ is not a well-defined notion. is closer than to , and is even closer
to than . We are merely trying to get an idea for what is happening graphically: if
we mentally ‘zoom’ in near the point , we want our linear approximation to
be as close as possible to the function we are approximating, in this case
.
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Now what’s so special about the linear approximation above? What is the slope of
that line? It looks like the slope of best linear approximation of the at is . This is no
coincidence! The best linear approximation of a differential function at is precisely
the tangent line to the curve at ! We have just found our last important property of
our best linear approximation at : The slope of the best linear approximation of a
function at is the slope of the line tangent to at , i.e. the derivative of evaluated at
()!
Given a function, a linear approximation (at ) is a fancy phrase for something you already know:
The line tangent to the function (at ).
Note that is just the line tangent to at .
Notice how the linear approximation (the purple line) tangentially hugs (the green line) at the value!