Math 141, Problem Set #9: Solutions to Additional Problems
A. Dierentiate arctan x + arccotx and explain why your answer makes
sense (in the same vein as our discussion of arccos x +arcsin x in class).
d
d
1
1
= dx arctan x+ dx arccotx = ( 1+x2 )+( 1+x2 )
[Pay Jeff $1.]
Section 4.7: Antiderivatives
Definition: A function F is called an antiderivative of f on
the set S if F (x) = f(x) for all x in S (except possibly at
endpoints of S).
If S isnt specified, its assumed to be the domain of f.
Example:
On the
Section 4.6: Newtons Method (concluded)
Recall the iterative formula underlying Newtons method
for obtaining better and better approximate solutions to the
equation f(x) = x:
xn+1 = xn f(xn) / f (xn).
We saw last time that the method can fail, and fail ba
[Collect summaries of sections 4.6; return HW. Pay
Michael $1.]
Section 4.5: Optimization problems (concluded)
In cases where there are two variables satisfying a
constraint, we dont need to solve for one of them in terms
of the other; we can find critica
[Collect summaries of sections 4.4 and 4.5; return HW.]
Section 4.5: Optimization problems
Basic problem: what values of x lead to the largest or
smallest possible values of f(x)?
A global maximum must be either an endpoint or a local
maximum, and a local
Section 4.3: Derivatives and the shapes of graphs
(concluded)
Concavity Test:
(a) If f (x) > 0 for all x in I, f is concave up on I.
(Is the converse true?
.?.
.?.
No: f(x) = x4 is concave up on [1,1], but f (x) is not
positive for all x in [1,1].)
(b) If
[Collect summaries of section 4.3; collect homework and
time-sheets; hand out time-sheets.]
Section 4.3: Derivatives and the shapes of graphs
Main ideas?
.?.
.?.
The geometric meaning of the first derivative
Increasing/decreasing test: Suppose f is contin
[Collect summaries of section 4.2]
Section 4.2: The Mean Value Theorem
Main ideas?
.?.
.?.
Mean Value Theorem: [Put this on the board]
Let f be a function on the interval [a,b] (with a < b) that
satisfies
1. f is continuous on the closed interval [a,b]
2.
[Collect summaries of section 4.1]
Section 4.1:
Key concepts?
.?.
.?.
Absolute (or global) vs. relative (or local) maxima and
minima, critical numbers
A function f with domain D has a global maximum at c if
f(c) f(x) for all x in the domain of f. We call
Prologue to section 3.7: Limits of ratios
How can we determine the limiting behavior of f(x)/g(x) in
terms of the limiting behavior of f(x) and the limiting
behavior of g(x)?
For simplicity, lets assume f(x) and g(x) are positive for all
x. Also, well loo
[Collect summaries of section 3.5; hand out time-sheets]
[Pay Flavio $1]
Section 3.4: Exponential growth and decay (concluded)
Interest
If you put $100 in the bank and its compounded yearly
with 12% annual interest, after 1 year you get your original
$100
[Collect summaries of section 3.4]
[Mention that Jef found the hidden note in the solution to
assignment #7]
Section 3.4: Exponential growth and decay
Let f(t) = # of bacteria in a colony at time t (where t is
measured in seconds).
Assume the colony grows
For Wednesday: Do the true/false questions for chapter 4.
Also, does there exist a twice-differentiable function f on R
satisfying f (x) < 0 and f (x) > 0 for all x?
[Collect homework; hand out practice exam]
Section 4.7: Antiderivatives (continued)
Recap
[Hand out solutions to practice exam]
Review of what weve seen so far:
The function |x|, defined on all of R, has an antiderivative
on R.
However, the Heaviside function, which is also defined on
all of R, does NOT have an antiderivative on R; it only has
Math 141, Problem Set #7: Solutions to Additional Problems
A. Derive the Quotient Rule from the Product Rule, the basic Power Rule
(the version on page 96, not the one on page 117!), and the Chain Rule.
(Hint: f (x)/g (x) can be written as f (x) times 1/g
Math 141, Problem Set #8: Solutions to Additional Problems
A. Sketch the curve given by f (x) =
x2 .
This is the same as the absolute value function: Its graph is the upper
half of the line of slope 1 through the origin together with the upper
half of the
Math 141, Problem Set #10: Solutions to Additional Problems
A. True or false? (and why?): If f is continuous on a closed interval
[a, b], then f attains a local maximum value f (c) and a local minimum
value f (d) at some numbers c and d in [a, b].
False.
Math 141, Problem Set #12: Solutions to Additional Problems
A. For both (a) and (b), use the First Derivative Test to conrm that the
relevant critical point is indeed a local minimum.
(a) Find the point on the graph of y = |x| that is closest to the point
Math 141, Problem Set #11: Solutions to Additional Problems
A. Let f (x) =
(x2 1)2 .
(a) Analyze the behavior of f in the vicinity of x = 0 using the First
Derivative Test.
Write f (x) = |x2 1| = h(g (x) where g (x) = x2 1 and h(y ) = |y |.
By the chain r
Math 141, Problem Set #6: Solutions to Additional Problems
A. Consider the function
x + 1 for x 1
for 1 < x < 0
f (x) = 0
2
x
for x 0.
Compute the derivative of f , being careful to say where it is undened.
The functions x + 1, 0, and x2 , being polynomia
Math 141, Problem Set #5: Solutions to Additional Problems
A. If we know limxa f (x) = and limxa g (x) = , what if anything
can we conclude about limxa f (x)g (x)? Explain. (Epsilons and deltas
are not required.)
When f (x) is really negative and g (x) is
Math 141, Problem Set #4: Solutions to additional problems
A. Suppose f (x) is continuous at x = a and g (x) is discontinuous at x = a.
Either prove that (f + g )(x) must be discontinuous at x = a or give an
example to show that it is possible for (f + g
Math 141, Problem Set #3: Solutions to additional problems
A. Prove that |ab| = |a|b| for all real numbers a, b.
There are four cases to consider.
a 0, b 0: ab 0, so |ab| = ab = |a|b|.
a 0, b 0: ab 0, so |ab| = ab = (a)(b) = |a|b|.
a 0, b 0: ab 0, so |ab|
Math 141, Problem Set #2: Solutions to additional problems
A. Let f (x) = 1/(1 x). Graph the functions f , f f , and f f f (be
careful about graphing the last of these!).
The graph of f (x) = 1/(1 x) is the graph of a hyperbola with asymptotes x = 1 and y
Math 141, Problem Set #1: Solutions to additional problems
A. Let x, y , and z respectively denote Alices score on the homework,
midterm, and nal exam for Math 141, so that her score S for the
course as a whole is determined by the formula
S = max(0.3x +
Questions about the practice exam or anything else?
Some subtle pitfalls to avoid
Here are some things I wont ask you on the exam, but
which should serve the purpose of reminding you of the
importance of not jumping to conclusions.
Suppose f(x) is a diffe
[Collect summaries of section 3.3]
Section 3.3: Derivatives of logarithmic functions and
exponential functions
Main ideas?
.?.
.?.
The derivative of exp(x) is exp(x).
The derivative of ln x is 1/x.
Basic Facts:
(1). (d/dx) loga x = (1/(loge a) (1/x) = (1
[Show handwriting samples for summaries of section 1.1.]
[Hand out solutions to practice midterm exam.]
[Collect summaries of section 3.1 and 3.2]
Section 3.2: Inverse functions
A function y=f(x) is called a one-to-one function if f(x1)
f(x2) whenever x1
Earlier this week I said there exists a nice differentiable
function f such that true-false quiz question 6 from Chapter
2 is only half-true for f, in the sense that the left hand side
(d/dx) f(x1/2)
is defined at x = 0 but the right hand side
f (x1/2)/(2