Warm Up

1. Warm Up Let f(x) = 1 – 2x2 1) Use Newtons Method to approximate the positive solution to f(x) = 0. (Use x1 = 1 and 3 iterations) 2) Determine the value of f(1) and f(1.3). What is the value of f(1.3) – f(1)? 3) Write an equation of the line tangent to f(x) at x = 1. Use the tangent line to approximate f(1.3).

2. Differentials

3. If y = f(x) is a function that is differentiable, then the differential of x (denoted dx) is a non-zero real number and the differentialof y (denoted dy) is defined to be dy = f’(x)dx The differential is used to approximate the change in y.

4. Find the differential dy.y = xcosx

5. Let f(x) = 1 – 2x2 Graph f(x) on the interval [.8, 1.5] using an x-scale of 0.1 and y-scale 0.5 Plot the point (1, f(1)). Using x = 0.3, show the change in the y-coordinates (y) on the graph. Graph the tangent line to f(x) at the point where x = 1. Using dx = 0.3, show the differential dy on the graph.

6. Approximations using differentials. dyis the change in y along the tangent line and is the approximation. If dx = Δx (and is small), then dy is a reasonable approximation for Δy (the actual change in the y coordinate). f(x) Tangent line Δy change in y on the function Change in y on the tangent line dy Δx

7. Given: y = ½x4 – 3x, x = -1, dx = Δx = 0.1 Evaluate dy (approximation) Evaluate Δy(actual change in y)

8. Let Determine dy. Evaluate dy (estimated change) for x = -2 and dx = 0.1 Evaluate y (actual change) for x = -2 and x = 0.1

9. Use differentials to approximate SOLUTION x = 16 and dx = 0.5 Determine dy. The approximate value …f(x + x) = f(x) + dy.