Calculus
Project #25 Finding Slopes Algebraically, a General Approach
A
tangent line to a curve is the best linear approximation to the curve near
the point of tangency.
Suppose
that the line y = m1(x-a) + b1 is tangent
to the graph of y = f(x) at the point (a, f(a)),
and that
the line y = m2(x-a) + b2 is tangent to the
graph of y = g(x) at the point
(a, g(a)).
-
What's
f(a)?
-
What's
g(a)?
-
What's
f'(a)?
-
What's
g'(a)?
If
we write L1(x) = m1(x-a) + b1 and
L2(x)
= m2(x-a) + b2, then L1 should
be a good approximation to f near x=a and L2 should
be a good approximation to g near x=a.
-
Find
a good approximation to the function h near x=a, where h(x) =
c*f(x).
-
What's
h(a)?
-
What's
h'(a)?
-
Find
a good approximation to the function F near x=a, where F(x) =
f(x) + g(x).
-
What's
F(a)?
-
What's
F'(a)?
-
Find
a good approximation to the function G near x=a, where G(x) =
f(x) - g(x).
-
What's
G(a)?
-
What's
G'(a)?
-
Find
a good approximation to the function H near x=a, where H(x) =
f(x) * g(x).
-
The
most obvious answer to the last question isn't a linear function. Find
a linear approximation to your approximation. One way to do this is to
assume your approximation has a formula of the form m*(x-a)+b. Set
this equal to your original approximation. Then (1) set
x = a and solve for b, (2) substitute
this value for b, (3) solve the
resulting equation for m, and (4) set x = a
to find a constant value for m.
-
What's
H(a)?
-
What's
H'(a)?
You're now an expert on the product
rule.
-
Find
a good approximation to the function K near x=a, where K(x) =
f(x)/g(x).
-
The
most obvious answer to the last question isn't a linear function. Find
a linear approximation to your approximation.
-
What's
K(a)?
-
What's
K'(a)?
You're now an expert on the quotient
rule.
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