Miss Battaglia BC Calculus. Let y=f(x) represent a functions that is differentiable on an open...

3-9 Continued & ReviewMiss Battaglia

BC Calculus

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!

Definition of Differentials

dyΔy

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

dyΔy

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)dy

Use differentials to approximate

Approximating Function Values

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?

Optimization

Rectilinear Motion

Rectilinear Motion

Example: s(t) = 2t3 – 21t2 + 60t + 3, 0 < t < 8

Describe the motion of the particle with a calculator.

Rectilinear Motion

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Classwork/Homework