Miss Battaglia
AB/BC Calculus

http://www.math.hmc.edu/calculus/tutorials
/tangent_line/
y = f(c) + f’(c)(x-c)
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.
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
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)dx
Use differentials to approximate
Using f(x) =
x
f(x) + f’(x)dx
16.5
Read 3.9 Page 240 #7, 11, 13, 15,
17, 27, 30, 43, 44, 53-56