Math 114, Calculus I, Laboratory 10:
Inverse Trigonometric Functions, Hyperbolic Functions and Extreme Values
1. Each of the six trigonometric functions (sin, cos, tan, cot, sec, csc) are periodic functions,
and so are clearly not one-to-one. However we can restrict their domains in different
ways so that the resulting function is one-to-one. See the second paragraph in section 3.6
for how this can be done with the function f ( x)  sin( x) . You know that a one-to-one
function possesses an inverse. So upon suitable restriction of domain, each of the six
trigonometric functions has an inverse (sin-1 or arcsin are the names given to the inverse
function for the "restricted" sine function). If we let y= f 1 ( x) , then a fundamental
connection between a function and its inverse enables us to write f ( y )  x . We know
that y is itself a function of x, so a straightforward implicit differentiation with respect to
x of the equation f ( y )  x will yield f ( y ) y   1 . We can solve this equation for y  and
obtain an expression for the derivative of each of the inverse trigonometric functions in
this way. To obtain an expression in x for the derivative you can stand on your
understanding of triangle trigonometry. (See how Figure 9 on page 231 illustrates with a
triangle the equation tan( y )  x .) Section 3.6 demonstrates the derivatives of
sin 1 ( x) and tan 1 ( x) found at the top of page 233.
SUBMIT: (a) your solutions of problems 25, 27, 28 and 29 on page 234, (b) a graph of
g ( x)  cos 1 ( x) and its derivative on the same coordinate system and (c) a few sentences
explaining the connections between the graphs. (We have asked for explanations of this
type several times in the course, so your articulation of what you mean should be clear.)
2. The last paragraph on page 236 states: "The derivatives of the hyperbolic functions are
easily computed." Convince yourself of this.
SUBMIT: using the definitions of the hyperbolic functions on page 235 and your
knowledge of the rules of differentiation demonstrate each of the differentiation formulas
listed in box (1) on the top of page 237.
3. Read Fermat's Theorem on page 257, and look at problem 65 on page 261. Note that
density  is mass/volume and the mass of the water is given as 1 kg.
SUBMIT: (a) a graph of  
1
over the domain that is appropriate for this problem
V (t )
situation, (b) a localized graph showing the temperature at which water has its maximum
density accurate to within 10 11 kg/cc (really (c) and use calculus to determine the
temperature at which water has its maximum density. (d) How does this result illustrate
Fermat's Theorem?
NOTE: You might have to drag your graphs out WIDE to see the numbers on the vertical
axis. Before you print, in your plot command use the option axesfont=[TIMES,BOLD,6].
This smaller 6 point font should help with the numbers on the vertical axis.
4. Read Examples 7-9 and the two paragraphs that follow on page 258.
SUBMIT: An explanation of what the book means by the parenthetical comment: "In
other words, the converse of Fermat's Theorem is false in general."
Historical Note on Fermat's better know theorem (Fermat's Last Theorem)
Pierre de Fermat (1601-1665) was a lawyer and amateur mathematician. In about 1637, he annotated his copy
(now lost) of Bachet's translation of Diophantus' Arithmetika with the following statement:
Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in
infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem
mirabilem sane detexi. Hanc marginis exiguitas non caparet.
In English, and using modern terminology, the paragraph above reads as:
There are no positive integers such that x^n + y^n = z^n for n > 2 . I've found a remarkable proof of this
fact, but there is not enough space in the margin [of the book] to write it.
Fermat never published a proof of this statement. It became to be known as Fermat's Last Theorem (FLT) not
because it was his last piece of work, but because it is the last remaining statement in the posthumous list of
Fermat's works that needed to be proven or independently verified. All others have either been shown to be true
or disproven long ago.
Andrew Wiles, a Princeton mathematician, recently proved Fermat's Last Theorem in 1994. So mathematician
have worked on this one problem for 357 years! Doesn't this put that implicit differentiation problem you
worked on for two hours in a different perspective? For more information about this famous conjecture now a
theorem (i.e. finally proven), ask the Mathematics Department secretary for the NOVA video "The Proof".