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3-9 Continued & Review

3-9 Continued & Review. Miss Battaglia BC Calculus. Definition of Differentials. Let y=f(x) represent a functions that is differentiable on an open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted dy ) is

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3-9 Continued & Review

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  1. 3-9 Continued & Review Miss Battaglia BC Calculus

  2. Definition of Differentials Let y=f(x) represent a functions that is differentiable on an open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differential of y(denoted dy) is dy = f’(x) dx -dy is the change in y -dx is the change in x -delta y pick a point close to point as limit approaches 0 dy/dx is the change in y over change in x! Δy dy

  3. 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. Δy dy

  4. Error Propagation The measured value x is used to compute another value f(x), the difference between f(x+Δx) and f(x) is the propagated error. f(x + Δx) – f(x) = Δy Measurement Error Propagated Error MeasuredValue Exact Value

  5. Estimation of Error The measurement radius of a ball bearing is 0.7 in. If the measurement is correct to within 0.01 in, estimate the propagated error in the volume V of the ball bearing.

  6. Differential Formulas Each of the differential rules from Chapter 2 can be written in differential form. Let u and v be differentiable functions of x. Constant multiple: d[cu] = c du Sum or difference: d[u + v] = du + dv Product: d[uv] = udv + vdu Quotient: d[u/v] =

  7. Finding Differentials

  8. Finding the Differential of a Composite Function y = f(x) = sin 3x

  9. Finding the Differential of a Composite Function y = f(x) = (x2 + 1)1/2

  10. Differentials can be used to approximate function values. To do this for the function given by y=f(x), use the formula f(x + Δx) = f(x) + dy = f(x) + f’(x)dy

  11. Approximating Function Values Use differentials to approximate

  12. Optimization A window is being built and the bottom is a rectangle and the top is a semicircle.  If there is 12 meters of framing materials what must the dimensions of the window be to let in the most light?

  13. Rectilinear Motion

  14. Rectilinear Motion

  15. Rectilinear Motion Example: s(t) = 2t3 – 21t2 + 60t + 3, 0 < t < 8 • Describe the motion of the particle with a calculator.

  16. Classwork/Homework Take home quiz

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