18: we need the equation of a tangent line, and it's
better left in the form $y-(-3)=4(x-5)$ or $y=4(x-5)-3$, than "simplified" (which
obscures the point of tangency, i.e. $x=5$, and the function
value there, of $y=-3$).
19: Because the tangent line matches the function at the point
in terms of function value and slope, we can simply evaluate
the tangent function.
33: I suspect that this would have been more difficult had
the answer not been in the back! If it was easy for you,
however, that's great. (Teri?)
We'll take time to review today: you ask the questions. When
you're done, we'll move along.
Last time we concluded our discussion of Section 2.2 -- the derivative as a function
-- with some examples. Do you have any particular questions before we
move along?
If not, we'll start into Section 2.3 by using the definition of
the derivative to prove several of the formulas in this section. We've
already proven the first two rules (constant functions $f(x)=c$ and the
linear function $f(x)=x$).
Rules that help us avoid having to use the definition each time (with Proofs):
The sum rule
works as we'd hope: "The derivative of a sum
is the sum of the derivatives."
The constant multiple rule (makes sense)
The product rule (has a tricky bit)
The power rule (via dominoes -- i.e., mathematical
induction)
NOTE: at this point we have all the rules necessary
to differentiate all polynomials, without needing to
resort to the definition!
The derivative of the monomial $a_nx^n$ is $na_nx^{n-1}$.
And a polynomial is just a sum of these. So
$s'(t)=(at^2+bt+c)'=2at+b$
and
$s''(t)=(2at+b)'=2a$
Other rules:
The quotient rule!
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