Midterm 1 Flashcards
What are the rules of inequalities?
- a < b ==> a + c < b + c
- a < b ==> a - c < b - c
- a < b and c > 0 ==> ac < bc
- a < b and c < 0 ==> ac > bc
- a > 0 ==> 1/a > 0
- 0 < a < b ==> 1/b < 1/a
What are the properties of absolute values?
- |-a| = |a|
- |ab| = |a||b|
- |a +- b| <= |a| + |b|
What are the equations and inequalities involving absolute values?
- |x| = D <==> x = -D or x = D
- |x| < D <==> -D < x < D
- |x| > D <==> x < -D or x > D
- |x - a| = D <==> x = a - D or x = a + D
- |x - a| < D <==> a - D < x < a + D
- |x - a | > D <==> x < a - D or x > a + D
What does it mean for a function to be even or odd?
- we say that a function is even if f(-x) = f(x) for all x in D
- we say that a function is odd if f(-x) = -f(x) for all x in D
What must be true for a function f to have a limit L at x = a?
If and only if it has a left and right limit at a and both of these limits equal L
What is the limit of a constant function? What are the hypotheses that must be true in order to make that limit true?
If f is the constant function f(x) = k, then the limit as x approaches a of f(x) is k. This assumes that f(x) is defined for some numbers around a even if it isn’t defined at a
What is the limit of f(x) = x? What are the hypotheses that must be true in order to make that limit true?
The limit as x approaches a of f(x) is a. This assumes that f(x) is defined for some numbers around a even if it isn’t defined at a
What are the algebraic rules for limits? What must be true in order to apply the algebraic limit rules?
Assume that the limit as x approaches a of f(x) = L and the limit as x approaches a of g(x) = M
1. The limit as x approaches a of f(x) + g(x) = L + M
2. The limit as x approaches a of f(x) - g(x) = L - M
3. The limit as x approaches a of f(x) * g(x) = L * M
4. The limit as x approaches a of k * f(x) = k * L
5. The limit as x approaches a of f(x)/g(x) = L/M as long as M != 0
6. The limit as x approaches a of f(x)^m/n = L^m/n as long as m is an integer and n is a positive integer and L >= 0 if n is even, and L != 0 if m < 0
7. if f(x) <= g(x) on some open interval containing a, then L <= M
8. the limit as x approaches a of f(g(x)) = f(M) as long as f is continuous at M
You must know that L and M exist in order to apply the ALR.
What are the limit rules for polynomials? What must be true in order to apply this theorem?
- the limit as x approaches a of P(x) = P(a)
- the limit as x approaches a of P(x)/Q(x) = P(a)/Q(a) as long as Q(a) != 0
What is the squeeze theorem?
Assume that f(x) <= g(x) <= h(x) on an open interval containing a, except possibly at a.
If the limit as x approaches a of f(x) is equal to the limit as x approaches a of h(x) = L, then the limit as x approaches a of g(x) = L
What must be true for a function to be continuous at c?
If the function’s domain contains an open interval containing c, and the limit as x approaches c of f(x) = f(c)
(The limit of f exists at c, the domain of f contains c, and the limit of f as x approaches c equals f(c))
What must be true for a function to be continuous on an interval?
It must be continuous at every point in that interval. At right end points, we only require left continuity and vice-versa
What must be true for a function to be continuous on its domain?
It must be continuous at every point in its domain. At right end points, we only require left continuity and vice-versa
What are the algebraic rules for combining continuous functions? What must be assumed to apply these rules?
Assuming that the functions f and g are defined on an interval containing c, and that f and g are both continuous at c.
1. f + g and f - g are continuous at c
2. fg is continuous at c
3. kf is continuous at c for any constant k
4. f/g is continuous at c so long as g(c) != 0
5. (f(x))^1/n is continuous at c provided that f(c) >= 0 if n is even
6. f(g(x)) is continuous at c if g is continuous at c and f is continuous at the limit as x approaches c of g(x)
What is a continuous extension?
Let f be a function such that the limit as f(x) = L exists but a is not in the domain of f. In that case, we can define a new function F(x) such that F(a) is defined as L
What is a removable discontinuity?
If f is a function such that the limit as x approaches a of f(x) = L exists, f is defined at a, but f is not continuous at a, then we say that f has a removable discontinuity at a. We can define a new function F(x) such that F(a) is defined as L
What is the Max/Min theorem? What must be true in order to apply the max/min theorem?
If a function is defined and continuous on a closed interval, there are minimums and maximums in that interval
The interval in question must be closed and finite
The function must be defined and continuous on that interval
What is the intermediate value theorem?
If a function is defined and continuous on a closed interval [a,b], then for any number between f(a) and f(b) there is a c in [a, b] such that f(c) = s.
The interval in question must be closed
The function must be defined and continuous on the interval
What must be true for a function to be differentiable at x?
The limit of the Newton Quotient must exist (the derivative must exist)
What can we assume if we know that f is differentiable at x?
f is also continuous at x (but continuity does not imply differentiability)
What are the rules for combining the derivatives of functions?
Assuming f and g are differentiable at x,
1. (f + g)’(x) = f’(x) + g’(x)
2. (f - g)’(x) = f’(x) - g’(x)
3. (Cf)’(x) = Cf’(x)
4. (fg)’(x) = f’(x)g(x) + f(x)g’(x)
5. (f/g)’(x) = [f’(x)g(x) - f(x)g’(x)]/g(x)^2 as long as g(x) != 0
What is the chain rule?
If f is differentiable at g(x) and g(x) is differentiable at x, then f(g(x)) is differentiable at x and
f(g(x))’ = f’(g(x))g’(x)
What is a differential equation?
An equation which involves the first or higher order derivatives of a function. A solution to the differential equation is a function which satisfies the equation
What is the mean value theorem?
if f is continuous on [a,b] and differentiable on (a,b), then there is a c in (a,b) such that the slope of the line between f(a) and f(b) equals f’(c)
What is a critical point?
A point x where f’(x) = 0
What is the derivative of a maximum or minimum?
By theorem 14, if f achieves a maximum or minimum value at c in (a,b) and f’(c) exists, then f’(c) = 0
What is Rolle’s theorem?
if g is continuous on [a,b], differentiable on (a,b), and g(a) = g(b), then there is a c in (a,b) such that g’(c) = 0
