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ADVANCED CALCULUS PRACTICE PROBLEMS.pdf
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ADVANCED CALCULUS PRACTICE PROBLEMSJAMES KEESLING
The problems that follow illustrate the methods covered in class. They are typical of
the types of problems that will be on the tests.
1. Riemann Integration
Problem 1. Let f : R → R be a function. State the definition of the derivative of f at
a point a ∈ R.
Problem 2. Let f : [a, b] → R be a bounded function. State when the Riemann integral
Rb Rb
of f (x) over [a, b], a f (x)dx, exists. What is the value of a f (x)dx when it does exist.
Rb
Problem 3. Suppose that f : [a, b] → R is monotone. Show that a f (x)dx exists.
n o
x ∈ [0, 1] x = ∞ an
P
Problem 4. Let C = n=1 3n , an ∈ {0, 2} be the Cantor set. Let
P∞ an /2 P∞ an
f : C → [0, 1] be defined by f x) = n=1 2n where x = n=1 3n is an element of the
Cantor set. This is called the Cantor ternary function. This function can be extended
so that f : [0, 1] → [0, 1] by making f be constant on the intervals that are complementary
R1
to the Cantor set. Compute the Riemann integral 0 f (x)dx of this function f (x).
Problem 5. Let {ri }∞ i=1 be an enumeration of the rational numbers in (0, 1). Define
f : [0, 1] → [0, 1] by the formula f (x) = ri <x 21i for 0 < x ≤ 1 and f (0) = 0. Show that f
P
R1
is an increasing function. Determine the value of 0 f (x)dx.
Pn 2
R1
Problem 6. Determine a formula for i=1 i . Use this to determine 0 x2 dx using the
Riemann sum definition of the integral.
Problem 7. Suppose that f (x) is piecewise monotone on [a, b]. By that we mean that
[a, b] = [x0 = a, x1 ] ∪ [x1 , x2 ] ∪ · · · [xn−1 , xn = b] with f (x) monotone on [xi , xi+1 ] for
Rb
0 ≤ i < n. Show that the Riemann integral for f (x), a f (x)dx, exists.
Problem 8. State and prove the Fundamental Theorem of Calculus.
1
, 2 JAMES KEESLING
Problem 9. Suppose that f (x) is continuous on [a, b]. Show that the Riemann integral
Rb
for f (x), a f (x)dx, exists.
Problem 10. Explain how Romberg integration works. Be able to use the TI-Nspire CX
CAS program to determine the Romberg estimate of integral.
2. Definition of ln(x) and exp(x)
Rx
Problem 11. Define ln(x) = 1 1t dt. Show that d ln(x)
dx ≡ x1 . Show that ln(x) is strictly
monotone increasing. Show that ln(x · y) = ln(x) + ln(y) for all x, y > 0. Show that
limx→∞ ln(x) = ∞ and that limx→0+ ln(x) = −∞
Problem 12. Define exp(x) = y where ln(y) = x. Show that exp(x + y) = exp(x) · exp(y),
limx→−∞ = 0, and limx→∞ = ∞. Show also that d exp(x)
dx ≡ exp(x).
Problem 13. Let a > 0. Define ab = exp(b · ln(a)). Calculate the following using this
definition.
d x
a
dx
d x
x
dx
lim (1 + 1/x)x
x→+∞
Problem 16. Define e > 0 by ln(e) = 1. Show that exp(x) ≡ ex for this e.
ln(x)
Problem 17. Define logb (x) = y such that by = x. Show that logb (x) ≡ ln(b) .
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