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 be the point where the tangent touches the graph, be the tangent, and be the function.
On the picture, the point is the coordinate of the vertical line.
Draw a vertical line from to the axis.
Draw horizontal lines from , , and the intersection of the vertical line with the tangent line.
At this stage, you should have three points on the axis: , , . Label them.
- Which of these points can you use to represent the estimate, or linear approximation, of near ?
- 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, is shown. The tangent line at is . The trace object is at the point (0.2, 5.488).
If you draw a vertical line from the axis through this point, you will get the point .
The distance or 7.04.
- What numerical value represents the linear approximation of near ?
- What numerical value gives the error associated with this linear approximation?
- What is the true value of ?
- Is this an underestimation or an overestimation?
Repeat the process above and complete the table below for the values given.
distance | linear approx.of | real valueof | error | underestimation/overestimation | ||
---|---|---|---|---|---|---|
- At what value(s) is the error less than ?
- 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 . Evaluate it at . 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
- What is ? What does this value represent?
- Calculate the error with this estimate.
Part 3 – Underestimates versus overestimates
Graph the function and place a tangent line .
- If you were to draw a point on the graph to the left of , is the approximation an overestimate or an underestimate?
- If you draw a point on the graph to the right of , 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 be for the linear approximation of at to be within 0.2 units of the true value of ?
Graph and with and the tangent line.
- How would you use the graphs to answer the question posed in this problem?
- How close to -1 must be for the linear approximation of at to be within 0.2 units of the true value of ?
Now we want to ask the same questions when the point of tangency is at .
- 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 is accurate to within 0.2 units of .