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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 / - PowerPoint PPT Presentation
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3.9 DifferentialsObjective: Understand the concept of a tangent line approximation and find the differential of a function
using differentiation formulas.
Miss BattagliaAB/BC Calculus
http://www.math.hmc.edu/calculus/tutorials/tangent_line/
y = f(c) + f’(c)(x-c)
Tangent Line Approximations
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.
Using a Tangent Line Approximation
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
Definition of Differentials
Let y=x2. Find dy when x=1 and dx=0.01. Compare this value with Δy for x=1 and Δx=0.01.
Comparing Δy and dy
dy
Δy
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
Error Propagation
ExactValue
Measured Value
Measurement Error
Propagated 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.
Estimation of Error
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 duSum or difference: d[u + v] = du + dvProduct: d[uv] = udv + vduQuotient: d[u/v] =
Differential Formulas
Function Derivative Differential
y=x2
y=2sinx
y=xcosx
y=1/x
Finding Differentials
y = f(x) = sin 3x
Finding the Differential of a Composite Function
y = f(x) = (x2 + 1)1/2
Finding the Differential of a Composite Function
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
Use differentials to approximate
Using f(x) = f(x) + f’(x)dx
Approximating Function Values
Read 3.9 Page 240 #7, 11, 13, 15, 17, 27, 30, 43, 44, 53-56
Classwork/Homework