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Miss Battaglia AB/BC Calculus

3.9 Differentials Objective: Understand the concept of a tangent line approximation and find the differential of a function using differentiation formulas. Miss Battaglia AB/BC Calculus. Tangent Line Approximations. http://www.math.hmc.edu/calculus/tutorials/tangent_line /

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Miss Battaglia AB/BC Calculus

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  1. 3.9 DifferentialsObjective: Understand the concept of a tangent line approximation and find the differential of a function using differentiation formulas. Miss Battaglia AB/BC Calculus

  2. Tangent Line Approximations • http://www.math.hmc.edu/calculus/tutorials/tangent_line/ y = f(c) + f’(c)(x-c)

  3. Using a Tangent Line Approximation Find the tangent line approximation of f(x)=1+sinx at the point (0,1). Then use a table to compare the y-values of the linear function with those of f(x) on an open interval containing x=0.

  4. Definition of Differentials Let y=f(x) represent a function 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

  5. 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

  6. 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

  7. 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.

  8. 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] =

  9. Finding Differentials

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

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

  12. 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)dx

  13. Approximating Function Values Use differentials to approximate Using f(x) = f(x) + f’(x)dx

  14. Classwork/Homework Read 3.9 Page 240 #7, 11, 13, 15, 17, 27, 30, 43, 44, 53-56

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