Prof. J. Beachy Review for Math 229, EXAM 2 10/23/2006
Exam 2 is scheduled for Friday, October 27, 2006. It will cover Sections 3.5–3.10 and 4.1–4.3.
You need to know these formulas: d
dx [f(x)g(x)] = f0(x)g(x) + f(x)g0(x)d
dx f(x)
g(x)=f0(x)g(x)−f(x)g0(x)
g(x)2
Section 3.5: d
dx (sin x) = cos xd
dx (cos x) = −sin xd
dx (tan x) = sec2xd
dx (sec x) = sec xtan x
Many of the limits reduce to lim
x→0
sin x
x= 1
Section 3.6: Chain rule: d
dx f(u(x)) = f0(u(x))u0(x)d
dx (u(x))n=n(u(x))n−1u0(x)
If y=f(u(x)), the chain rule can also be written as dy
dx =dy
du
du
dx
Section 3.7: Given an equation in the variables xand y, instead of solving for yin terms of xand then finding y0,
you can think of yas an implicit function of xand use the chain rule to differentiate both sides of the equation with
respect to x. This gives a way to solve for y0, but the answer is then typically given in terms of both xand y.
Section 3.8: Depending on the function, you can repeat the process of taking a derivative. Notation:
f00(x), f000 (x), etc, or d2y
dx2=d
dx dy
dx ,d3y
dx3=d
dx d2y
dx2, etc. If s=f(t) gives the position of an object as a
function of time, then the first derivative f0(t) gives its velocity, and the second derivative f00(t) gives its acceleration.
We can write v=f0(t), and a=f00(t), or v=ds
dt , and a=dv
dt =d2s
dt2.
Section 3.9: Here is the typical related rates problem. If yis a function of x, and xis a function of the time t, then
the chain rule says that dy
dt =dy
dx
dx
dt . Remember that dy
dt is the rate of change of ywith respect to time, and dx
dt
is the rate of change of xwith respect to time. If you know one of these two rates, and can calculate dy
dx , then you
can find the other rate. In some of the problems, when you find an equation relating xand y, it may be easier to
use implicit differentiation (with respect to time) to find an equation relating dy
dt and dx
dt , rather than first solving
for yin terms of x, and then differentiating.
Section 3.10: Using the way we have constructed tangent lines, if xis close to a, then the corresponding points on
the graph of y=f(x) are close to the tangent line. In this section, the author writes the tangent line at (a, f (a))
in the form L(x) = f(a) + f0(a)(x−a), and calls it the linearization of f(x) at a. When xis close to a, written
x≈a, then f(x) is approximated by the linearization at a, so f(x)≈f(a) + f0(a)(x−a). If you are only interested
in approximating the change in f(x), then all you need to use is the slope of the tangent line: if ∆x=x−ais
the change in x, and ∆y=f(x)−f(a) is the corresponding change in y, then ∆y≈f0(a)∆x. The author uses
differential notation, so that ∆y≈dy, where dy =f0(x)dx is the differential of y, in which dx plays the role of ∆x.
Section 4.1: Acritical number cof a function f(x) is a value of xfor which either f0(c) = 0 or f0(c) is undefined.
To search for local maximum and local minimum values of the function, we only need to look at the critical numbers
of the function. To find absolute maximum and absolute minimum values of a function on an interval [a,b], we must
look at f(c) for all critical numbers cand at f(a) and f(b).
Section 4.2: The Mean Value Theorem (p 235) is used primarily to prove other results. It says (roughly) that if
a function f(x) has a derivative, then in any interval [a,b] there is some number cwhere the instantaneous rate of
change f0(c) is the same as the average rate of change f(b)−f(a)
b−a. One application says that if two functions have
the same derivative, then their graphs are “parallel” (differ by a constant amount).
Section 4.3: If f0(x)>0, then f(x) is increasing (as xincreases); if f0(x)<0, then f(x) is decreasing. (Model:
linear functions.) Knowing the sign of f0(x) helps to graph y=f(x), and helps to decide when a critical number
produces a local maximum and when it produces a local minimum (see the First Derivative Test on p 241).
If f00(x)>0, then the graph of y=f(x) is concave up, since its slopes are increasing; if f00(x)<0, then the
graph of y=f(x) is concave down, since its slopes are decreasing. (Model: quadratic functions.) Knowing the sign
of f00(x) helps to graph y=f(x), and if cis a critical number with f00(c)>0, then f(c) is a local minimum since
f(x) is concave up; if f00(c)<0, then f(c) is a local maximum since f(x) is concave down (See the Second Derivative
Test on p 245.)