Aim: Differential? Isn’t that part of a car’s drive transmission?

1. Aim: Differential? Isn’t that part of a car’s drive transmission? Do Now: Find the equation of the tangent line for f(x) = 1 + sinx at (0, 1).

2. Linear Approximations y = x2 graph of function is approximated by a straight line. y = x2 = 2x - 1 y = x2 = 2x - 1

3. c, f(c) x x c Linear Approximations Can the graph of a function be approximated by a straight line? By restricting values of x to be close to c, the values of y of the tangent linecan be used as approximations of the values of f. (x, y) f as x c, the limit of s(x) or y is f(c) s equation of tangent line y2 – y1 = m(x2 – x1) - point slope y – f(c) = f’(c) (x – c) Equation of tangent line approximation y = f(c) + f’(c)(x – c)

4. = 1 + x Model Problem Find the tangent line approximation of at the point (0, 1). 1st derivative of f Equation of tangent line approximation y = f(c) + f’(c)(x – c) y = 1 + cos 0 (x – 0) y = 1 + 1x The closer x is to 0, the better the approximation.

5. Δy dy Differential derivative of y with respect to x also the ratio dy  dx is the slope of the tangent line When we talk only of dy or dx we talk differentials As Δx gets smaller and smaller, before it reaches 0,  approximates actual change approximation of Δy

6. Δy f’(c)Δx f(c + Δx) c, f(c) Δx f(c) c + Δx c Differential Approximations (c + Δx, f(c + Δx) dy the ratio dy  dx is the slope of the tangent line = dx When Δx is small, then Δy is also small and Δy = f(c + Δx)– f(c) and is approximated by f’(c)Δx.

7. Definition Let y = f(x) represent a function that is differentiable in an open interval containing x . The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx Differential When Δx is small, then Δy is also small and Δy = f(c + Δx)– f(c) and is approximated by f’(c)Δx. Δy = f(c + Δx)– f(c) actual change in y  f’(c)Δx approximate change in y Δy dy Δy  f’(c)Δx

8. Differential Definition Δxis an arbitrary increment of the independent variable x. dx is called the differential of the independent variable x, dx is equal to Δx. Δy is the actual change in the variable y as x changes from x to x + Δx; that is, Δy = f(x + Δx)– f(x) dy, called the differential of the dependent variable y, is defined by dy = f’(x)dx

9. actual change in y Δy = f(c + Δx)– f(c) Comparing Δy and dy Let y = x2. Find dy when x = 1 and dx = 0.01. Compare this value with Δy for x = 1 and Δx = 0.01. approximate change in y dy = f’(x)dx y = f(x) = x2 f’(x) = 2x dy = f’(1)(0.01) dy = 2(1)(0.01) dy = 2(0.01) = 0.02

10. dy = 2(0.01) = 0.02 Comparing Δy and dy Let y = x2. Find dy when x = 1 and dx = 0.01. Compare this value with Δy for x = 1 and Δx = 0.01. approximate change in y actual change in y dy = f’(x)dx Δy = f(c + Δx)– f(c) Δy = f(1 + 0.01)– f(1) Δy = f(1.01)– f(1) Δy = 1.012– 12 Δy = 0.0201 values become closer to each other when dx or Δx approaches 0

11. Δx = 0.01 Comparing Δy and dy Let y = x2. Find dy when x = 1 and dx = 0.01. Compare this value with Δy for x = 1 and Δx = 0.01. = 0.0201 = 0.02 1, 1

12. difference is propagated error Error Propagation estimations based on physical measurements A(r) = πr2 r = 7.2cm – exact measurement A(7.2) = π(7.2)2 = 162.860 7.21 cm 7.19 cm 7.18 cm A = 163.313 A = 162.408 A = 161.957

13. Measurement error Propagated error Exact value Measurement value Error Propagation Propagation error – when a measured value that has an error in measurement is used to compute another value. f( ) x + Δx  f(x) = Δy approximate change in y dy = f’(x)dx

14. approximate change in y dy = f’(x)dx Model Problem The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing. r = 0.7 measured radius possible error -0.01 <Δr < 0.01 approximate ΔV by dV substitute r and dr

15. Relative Error The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing.  4.29% relative error

16. Let u and v be differentiable functions of x. Definition The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx. Differential Formulas  dy = y’dx Liebniz notation

17. Differential Formulas Function Derivative Differential • y = x2 • y = 2sin x • y = xcosx • y = 1/x

18. Model Problem Find the differential of composite functions y = f(x) = sin 3x Original function y’ = f’(x) = 3cos 3x Apply Chain Rule dy = f’(x)dx = 3cos 3x dx Differential Form y = f(x) = (x2 + 1)1/2 Original function Apply Chain Rule Differential Form

19. Approximating Function Values Use differential to approximate x = 16 and dx = 0.5

20. Model Problem Use differential to approximate

21. Model Problem Find the equation of the tangent line T to the function f at the indicated point. Use this linear approximation to complete the table.

22. Model Problem The measurement of the side of a square is found to be 12 inches, with a possible error of 1/64 inch. Use differentials to approximate the possible propagated error in computing the area of the square.

23. Model Problem The measurement of the radius of the end of a log is found to be 14 inches, with a possible error of ¼ inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

24. Model Problem The radius of a sphere is claimed to be 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum possible error in calculating (a) the volume of the sphere, (b) the surface area of the sphere, and (c) the relative errors in parts (a) and (b).