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.

Given the function (in black), whose graph is below, what is the error of each of these various approximations (Lines A, B, C , and D ) of at ?

PIC

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
The ‘error of an approximation’ is the distance between the true solution and the approximation. (More Hints Available)
If you are approximating at , then the ‘true solution’ is the true value of at , AKA . (More Hints Available)
The approximation of the function at using a line is .(More Hints Available)
Error shouldn’t be negative. If you are finding the difference between two numbers and the difference is negative, then the absolute value of the difference would give the distance between the numbers.
Given the errors you found for at using each line (A,B,C, and D), which line(s) has (have) the least error at ?
A B C D

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.

PIC

Let’s ask another question: What are the errors of the approximations ‘near’ ?

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.

Typing ‘’ into a blank entry on the desmos calculator evaluates the line at . (More Hints Available)
Typing ‘’ would compute the error of the approximation of at using the line .

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

Given the errors you found for at and using the lines B,C, and G, which line(s) has (have) the least error ‘near’ ?
B C G

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 .