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!
Δy
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
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.
Measurement
Error
Propagated
Error
f(x + Δx) – f(x) = Δy
Exact
Value
Measured
Value
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:
Sum or difference:
Product:
Quotient:
d[cu] = c du
d[u + v] = du + dv
d[uv] = udv + vdu
d[u/v] = vdu - udv
v2
Function
Derivative
y=x2
dy
= 2x
dx
y=2sinx
y=xcosx
y=1/x
Differential
dy = 2xdx
y = f(x) = sin 3x
y = f(x) = (x2 + 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
16.5
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) = t3 – 6t2, 0 < t < 8
Position versus time curve
s(t)
s(t) = t3 – 6t2
Velocity versus time curve
v(t)
v(t) =
Acceleration versus time curve
a(t)
a(t) =
Example: s(t) = 2t3 – 21t2 + 60t + 3, 0 < t < 8

Describe the motion of the particle with a
calculator.
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