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