4.4: Linear Approximations
This activity is intended to supplement Calculus, Chapter 3, Lesson 8.
Part 1 – Introduction
Linear approximation uses a tangent line to estimate values of a function near the point of tangency. For this reason, linear approximation is also referred to as tangent line approximation.
On the graph to the right, let
On the picture, the point
Draw a vertical line from
Draw horizontal lines from
At this stage, you should have three points on the
 Which of these points can you use to represent the estimate, or linear approximation, of
f(x) neara ?  How can you use these labels to represent the error associated with this estimate?
 Is this estimate an overestimate or an underestimate? Explain.
Part 2 – Investigating linear approximation
On the graph at the right,
If you draw a vertical line from the
The distance
 What numerical value represents the linear approximation of
f1(q) neara=−1 ?  What numerical value gives the error associated with this linear approximation?
 What is the true value of
f1(q) ?  Is this an underestimation or an overestimation?
Repeat the process above and complete the table below for the

distance 
linear approx.of 
real valueof 
error  underestimation/overestimation  








 At what
x− value(s) is the error less than0.5 ?  What do you notice about graph of the function and the graph of the tangent line as you get close to the point of tangency?
 Based on your observations, explain why the relationship between a tangent and a graph at the point of tangency is often referred to as local linearization.
Typically, you will have a function but not a graph to find the linear approximation.
 Find the derivative of
f1(x)=x3−3x2−2x+6 . Evaluate it atx=−1 . This is the slope of the line. Use the slope and the point (1, 4) to get the equation of the line.  The tangent line
L(x)=  What is
L(−1.03) ? What does this value represent?  Calculate the error with this estimate.
Part 3 – Underestimates versus overestimates
Graph the function
 If you were to draw a point
p on the graph to the left ofa=1 , is the approximation an overestimate or an underestimate?  If you draw a point
p on the graph to the right ofa=1 , is the approximation an overestimate or an underestimate?  What is the significance of the point of tangency?
 Generalize your findings about when a linear approximation produces an overestimate and when it produces an underestimate.
Part 4 –Finding intervals of accuracy
How close to 1 must
Graph \begin{align*}f2 = f1 + 0.2\end{align*} and \begin{align*}f3 = f1  0.2\end{align*} with \begin{align*}f1\end{align*} and the tangent line.
 How would you use the graphs to answer the question posed in this problem?
 How close to 1 must \begin{align*}x\end{align*} be for the linear approximation of \begin{align*}f1(x) = x^3  3x^2  2x + 6\end{align*} at \begin{align*}a = 1\end{align*} to be within 0.2 units of the true value of \begin{align*}f1(x)\end{align*}?
Now we want to ask the same questions when the point of tangency is at \begin{align*}a = 1\end{align*}.
 How does this situation differ from the one we just had?
 Use graphical or algebraic methods to find an interval that ensures the linear approximation at \begin{align*}a = 1 \end{align*} is accurate to within 0.2 units of \begin{align*}f1(x)\end{align*}.
Notes/Highlights Having trouble? Report an issue.
Color  Highlighted Text  Notes  

Show More 