1. 6.6—Differentials: Linear Approximation Objectives: Use differentials to approximate the change of a function’s value Use linear approximation to estimate the value of a function

2. Warm up • Graph f(x) = x2 • In the same window, graph the tangent line to f at (1,1) • Zoom in three times at the point (1, 1) • Use the trace feature to compare the two graphs • As the x values get closer to 1, what can you say about the y-values?

3. DIFFERENTIALS • Answers the question “When x changes by a small amount, how much does the function’s value change?” • We can use differentials to estimate change and approximate values of a function near known x values • Application of the tangent line

4. Differentials

5. DEFINITION • this is the actual change in f. • is the differential and is an approximation of • When is very small, ≈ • is the small change in (differential of , ) • is the derivative, slope of the tangent line • DEFINITION OF THE DIFFERENTIAL OF y:

6. Find dy. Then find dy for the given values of xand Δx. 2.

7. Evaluate the differential 4. , x=4, Δx=.02 • 3.

8. Linear Approximations • Uses differentials to estimate a function’s VALUE for given x-values • Let f be a function whose derivative exists. For small nonzero values of Δx: • f(x+ Δx) ≈ f(x)+ dy = f(x) + f’(x)dx • FUN!!!! We don’t need calculators!

9. Using differentials, approximate

10. Approximate

11. Differential Applications—Estimation of Change • Make sure to note the variables used. The differential should be in terms of the independent variable. Example: If ,