Math 160 Fall 1999
EXTRA CREDIT PROBLEMS:
1. Although the definition of the limit in Section 2.2 gives us a good
intuitive feeling for what a limit is, it is not really precise. Criticize
the intuitive definition of a limit in Section 2.2 and say how you might
improve it.
2. Give an example of a function that has a discontinuity that is not
of the type that the book mentions (that is, not removable, jump or infinite).
For extra brain exercise, try to make the function continuous everywhere
except at the point that has the "new" type of discontinuity.
3. Show that any polynomial of odd degree must have at least one real
root (even if you know what the derivative is, do NOT use it).
4. Using properties inverse functions and exponential functions, derive
the properties of logarithms listed in Section 1.6 of the Stewart book.
5. Using the identity e^(ix)=cos(x)+isin(x) (with i the square root
of -1) and the identities from section 3.9, show that cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
and sin(x+y)=sin(x)cos(y)+cos(x)sin(y). Show a relationship between the
(ordinary) trigonometric functions and the hyperbolic trig functions.
6. Derive the derivative of the inverse hyperbolic cosecant function.
7. Prove or give a counterexample to the following statement. If the
tangent line to y=f(x) at the point (a,b) is y=mx+b (m is not 0), then
the tangent line to the inverse function of f(x) at (b,a) is y=(1/m)x-b/m.
8. Let f(x) be a continuous, odd function. Find f(0) and prove your
answer.
9. Let f(x) be a 1-1 function whose graph goes through the point (a,b).
Assume that the tangent line to y=f(x) at (a,b) is given by y=mx+c with
m not equal to 0. What is the relationship (if any) between the tangent
line to y=f(x) at (a,b) and the tangent line to the inverse function of
f(x) at (b,a)?
10. Consider problem #35 in section 4.7 of Stewart's book. Find the
angle of the slice cut out of the paper circle that makes the volume a
maximum.
11. Verify the following "shortcut" formula for limits at infinity.
First let f(x) be a polynomial of degree n (f(x)=ax^n+...+c) and g(x) be
a polynomial of degree m (g(x)=bx^m+...+d) (a and b are nonzero). Show
that the limit of f(x)/g(x) as x goes to infinity is:
0, if n is less than m,
a/b, if n=m,
infinity, if n is greater than m and (a/b)>0
minus infinity, if n is greater than m and (a/b)<0.
12. Give an example of a function that has a point that is both a local
extremum and a point of inflection. When you are done with this give an
example of a function that has a point that is both a local extremum and
a point of inflection AND the derivative is defined at this point (or prove
that it cannot be done).
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