Miss 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! dy Δy

Let y=x 2. Find dy when x=1 and dx=0.01. Compare this value with Δy for x=1 and Δx=0.01. 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 Exact Value 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.

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] =

FunctionDerivativeDifferential y=x 2 y=2sinx y=xcosx y=1/x

y = f(x) = sin 3x

y = f(x) = (x 2 + 1) 1/2

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

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?

Example: s(t) = 2t 3 – 21t t + 3, 0 < t < 8  Describe the motion of the particle with a calculator.

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