Midterm 1

Question 1

What are the rules of inequalities?

Answer 1

1. a < b ==> a + c < b + c
2. a < b ==> a - c < b - c
3. a < b and c > 0 ==> ac < bc
4. a < b and c < 0 ==> ac > bc
5. a > 0 ==> 1/a > 0
6. 0 < a < b ==> 1/b < 1/a

Question 2

What are the properties of absolute values?

Answer 2

1. |-a| = |a|
2. |ab| = |a||b|
3. |a +- b| <= |a| + |b|

Question 3

What are the equations and inequalities involving absolute values?

Answer 3

1. |x| = D <==> x = -D or x = D
2. |x| < D <==> -D < x < D
3. |x| > D <==> x < -D or x > D
4. |x - a| = D <==> x = a - D or x = a + D
5. |x - a| < D <==> a - D < x < a + D
6. |x - a | > D <==> x < a - D or x > a + D

Question 4

What does it mean for a function to be even or odd?

Answer 4

1. we say that a function is even if f(-x) = f(x) for all x in D
2. we say that a function is odd if f(-x) = -f(x) for all x in D

Question 5

What must be true for a function f to have a limit L at x = a?

Answer 5

If and only if it has a left and right limit at a and both of these limits equal L

Question 6

What is the limit of a constant function? What are the hypotheses that must be true in order to make that limit true?

Answer 6

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

Question 7

What is the limit of f(x) = x? What are the hypotheses that must be true in order to make that limit true?

Answer 7

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

Question 8

What are the algebraic rules for limits? What must be true in order to apply the algebraic limit rules?

Answer 8

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.

Question 9

What are the limit rules for polynomials? What must be true in order to apply this theorem?

Answer 9

1. the limit as x approaches a of P(x) = P(a)
2. the limit as x approaches a of P(x)/Q(x) = P(a)/Q(a) as long as Q(a) != 0

Question 10

What is the squeeze theorem?

Answer 10

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

Question 11

What must be true for a function to be continuous at c?

Answer 11

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))

Question 12

What must be true for a function to be continuous on an interval?

Answer 12

It must be continuous at every point in that interval. At right end points, we only require left continuity and vice-versa

Question 13

What must be true for a function to be continuous on its domain?

Answer 13

It must be continuous at every point in its domain. At right end points, we only require left continuity and vice-versa

Question 14

What are the algebraic rules for combining continuous functions? What must be assumed to apply these rules?

Answer 14

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)

Question 15

What is a continuous extension?

Answer 15

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

Question 16

What is a removable discontinuity?

Answer 16

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

Question 17

What is the Max/Min theorem? What must be true in order to apply the max/min theorem?

Answer 17

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

Question 18

What is the intermediate value theorem?

Answer 18

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

Question 19

What must be true for a function to be differentiable at x?

Answer 19

The limit of the Newton Quotient must exist (the derivative must exist)

Question 20

What can we assume if we know that f is differentiable at x?

Answer 20

f is also continuous at x (but continuity does not imply differentiability)

Question 21

What are the rules for combining the derivatives of functions?

Answer 21

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

Question 22

What is the chain rule?

Answer 22

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)

Question 23

What is a differential equation?

Answer 23

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

Question 24

What is the mean value theorem?

Answer 24

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)

Question 25

What is a critical point?

Answer 25

A point x where f’(x) = 0

Question 26

What is the derivative of a maximum or minimum?

Answer 26

By theorem 14, if f achieves a maximum or minimum value at c in (a,b) and f’(c) exists, then f’(c) = 0

Question 27

What is Rolle’s theorem?

Answer 27

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