Chapter 2

Question 1

Define a limit?

Answer 1

Suppose f(x) s defined when (x) is near the number (a). This means that f is defined on some open interval that contains (a), except possibly at (a) itself.

lim(x-a)f(x) = L
“The limit of f(x), as (x) approaches (a), equals L”

if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to (a) (on either side of a) but not equal to (a)

Question 2

Define one-sided limits?

Answer 2

lim(x-a^-)f(x) = L
and say that the LEFT-HANDED LIMIT OF F(X) AS (X) APPROACHES (A) (or as (x) approaches (a) from the left) is equal to L

if we can make the values of f(x) arbitrarily close to L by taking (x) to be sufficiently close to (a) and (x) less than (a)

same goes for a right handed-limit

Question 3

Define Infinite Limits?

Answer 3

Let f be a function defined on both sides of (a), except possibly at (a) itself. Then…

lim(x-a)f(x) = ∞
means that the values of f(x) can be made arbitrarily large (as large as we please) by taking (x) sufficiently close to (a), but not equal to (a)

Let f be defined on both sides of (a), except possibly at (a) itself. Then…

lim(x-a)f(x) = -∞
means that the values of f(x) can be made arbitrarily large negative by taking (x) sufficiently close to (a), but not equal to (a)

Question 4

what is a vertical asymptote?

Answer 4

the line x=a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true

lim(x-a)f(x) = ∞  lim(x-a^-)f(x) = ∞ lim(x-a^+) = ∞
lim(x-a)f(x) = -∞ lim(x-a^-)f(x) = -∞ lim(x-a^+) = -∞

Question 5

what are the limit laws?

Answer 5

1. The limit of a sum is the sum of the limits lim(x-a)[f(x) + g(x)] = lim(x-a)f(x) + lim(x-a)g(x)
2. The limit of a difference is the difference of the limits
   lim(x-a)[f(x) - g(x)] = lim(x-a)f(x) - lim(x-a)g(x)
3. The limit of a constant times a function is the constant times the limit of the function
   lim(x-a)[Cf(x)] = Clim(x-a)f(x)
4. The limit of a product is the product of the limits
   lim(x-a)[f(x)g(x)] = lim(x-a)f(x)lim(x-a)g(x)
5. The limit of a quotient is the quotient of the limits (provided that the limit of the
   denominator is not 0)
   lim(x-a)[f(x) / g(x)] = lim(x-a)f(x) / lim(x-a)g(x); g(x) cannot equal 0

Question 6

what is the direct substitution property?

Answer 6

If f is a polynomial or a rational and (a) is in the domain of f, then

Lim(x-a)f(x) = f(a)

Question 7

what is the theorem that applies to limits?

Answer 7

If f(x)

Question 8

what is the squeeze theorem?

Answer 8

It says that if g(x) is squeezed between f(x) and h(x) near (a), and if f and h have the same limit at (a), then g is forced to have the same limit L at (a)

If f(x) = g(x) = h(x) when (x) is near (a) (except possibly at (a)) and…

Lim(x-a)f(x) = Lim(x-a)h(x) = L then…

Lim(x-a)g(h) = L

Question 9

A number divided by 0 = ?

Answer 9

Undefined

Question 10

Zero divided by a number = ?

Answer 10

0

Question 11

The special cubic factors for addition are?

Answer 11

a3 + b3 = (a + b)(a2 – ab + b2)

Question 12

The special cubic factors for subtraction are?

Answer 12

a3 – b3 = (a – b)(a2 + ab + b2)

Question 13

Define continuity?

Answer 13

A function f is continuous at a number (a) if..

Lim(x-a)f(x) = f(a)

A continuous process is one that takes place gradually, without interruption or abrupt change

Question 14

what are the 4 things that determine if a function is continuous or not on a specific interval?

Answer 14

1. f(a) is defined (that is, (a) is in the domain of f)
2. Lim(x-a)f(x) exists
   3.Lim(x-a)f(x) = f(a)
3. Lim(x-a^+)f(x) = f(a) = Lim(x-a^-)f(x) = f(a)
   a function f is continuous at a number (a) and f is continuous from the left (a)

Question 15

Define discontinuity?

Answer 15

If f is defined near (a) (in other words, f is defined on an open interval containing (a), except perhaps at (a)), we say f is discontinuous at (a), if f is not continuous at (a)

Question 16

what is the Intermediate Value Theorem?

Answer 16

Suppose that f is continuous on a closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) cannot equal f(b). Then there exists a number (c) in (a,b) such that f(c) = N

Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values f(a) and f(b)

Question 17

What are limits to infinity: horizontal asymptotes?

Answer 17

Let f be a function defined on some interval (a, ∞). Then…

Lim(x-∞)f(x) = L

means that the values of f(x) can be made arbitrarily close to L by taking x-sufficiently large

Question 18

What are limits to infinity: horizontal asymptotes negative?

Answer 18

Let f be a function defined on some interval (-∞, a). Then..

Lim(x->-∞)f(x) = L

means that the values of f(x) can be made arbitrarily close to L by taking x-sufficiently large NEGATIVE

Question 19

what is a horizontal asymptote?

Answer 19

The line y=L is called a horizontal asymptote of the curve y=f(x) if either…

Lim(x->∞)f(x) = L or Lim(x->-∞)f(x) = L

Question 20

To find the horizontal asymptote of a rational function without a square root, you do what?

Answer 20

divide the numerator and the denominator by the highest power of x in the denominator

Question 21

To find the horizontal asymptote of a rational function that has a square root in the numerator, you do what?

Answer 21

divide the numbers under the square root, by the highest power of x underneath the square root. All the other numbers divide by the highest power of x in the entire function

Question 22

what is a tangent line?

Answer 22

The slope of a curve at a point. The tangent line to the curve y = f(x) at a point P(a, f(a)) is the line through P with slope

m = lim(x-a) f(x) - f(a) / x -a

provided the limits

Question 23

what is the instantaneous velocity?

Answer 23

v(a) at a time t=a to be the limit of these average velocities

v(a) = lim(h-0) f(a + h) - f(a) / h

Question 24

what are the two derivative definitions?

Answer 24

1. The derivative of a function f at a number (a), denoted by f’(a) is…..

f’(a) = Lim(h->0) f(a + h) - f(a) / h

2. f’(a) = Lim(x->a) f(x) - f(a) / x - a

Question 25

what are differentiation factors?

Answer 25

f'(x) = y' = dy/dx = df/dx = d/dxf(x) = Df(x) = Dx f(􏰨x)􏰩

Question 26

what determines if a function is differentiable?

Answer 26

a function is differentiable at (a) if f'(a) exists. It is differentiable on an open interval (a,b) or (a,∞) or (a, -∞) If it is differentiable at every number in the interval

Question 27

what determines if a function is non-differentiable?

Answer 27

1. If a a graph of a function has a "kink" or a "corner" in it the graph has no tangent at this point and f is not differentiable there 2. If the left and right limits are different, if f is not continuous at a, then f is not differentiable at a. So at any discontinuity (for instance, a jump discontinuity) f fails to be differentiable 3. A third possibility is that the curve has a vertical tangent line when x 􏱀 a

Question 28

what are higher derivatives? what is a seconds derivative of a function?

Answer 28

This new function f'' is called the second derivative of f because it is the derivative of the derivative of f

Question 29

what is the derivative of a constant function?

Answer 29

d/dx (c) = 0

Question 30

What is the derivative power rule?

Answer 30

d/dx (x^n) = n(x^n-1)

Question 31

what is the definition of the number e?

Answer 31

Lim(h->0) (e^h -1) / h = 1

Question 32

what is the derivative of the Natural exponential function (e)?

Answer 32

d/dx (e^x) = e^x

Question 33

what is a the product rule for differentiation?

Answer 33

fg = f'g + g'f

Question 34

what is the quotient rule for differentiation?

Answer 34

f/g = f'g - g'f / g^2

Question 35

Lim(x->0) cosx - 1/x

Answer 35

0

Question 36

The derivative of sin(x) is?

Answer 36

cos(x)

Question 37

The derivative of cos(x) is?

Answer 37

-sin(x)

Question 38

The derivative of tan(x) is ?

Answer 38

sec(x^2)

Question 39

The derivative of csc(x) is?

Answer 39

-csc(x)cot(x)

Question 40

The derivative of sec(x) is?

Answer 40

sec(x)tan(x)

Question 41

The derivative of cot(x) is?

Answer 41

-csc(x^2)

Question 42

What is the chain rule?

Answer 42

If t is differentiable at x and f is differentiable at t􏰨x􏰩, then the composite function F = f ◦􏱑 t defined by F(􏰨x)􏰩 =􏱀 f(􏰨g(x))􏰩􏰩 is differentiable at x and F􏰐' is given by the product F'(x) = f'(g(x)) ● g'(x)

Question 43

What is implicit differentiation?

Answer 43

This method consists of differentiating both sides of the equation with respect to (x) and then solving the resulting equation for y'

Question 44

what is the derivative of inverse sin(x) (sinx^-1)?

Answer 44

1/√(1-X^2)

Question 45

what is the derivative of inverse cos(x) (cosx^-1)?

Answer 45

-1/√(1-X^2)

Question 46

What is the derivative of inverse tan(x) (tanx^-1)?

Answer 46

1/√(1+X^2)

Question 47

What is the derivative of inverse cot(x) (cotx^-1)?

Answer 47

-1/√(1+X^2)

Question 48

What is the derivative of inverse sec(x) (secx^-1)?

Answer 48

1/(x)√(x^2-1)

Question 49

What is the derivative of inverse csc(x) (cscx^1)?

Answer 49

-1/(x)√(x^2-1)

Question 50

what is the derivative of a Logarithmic function?

Answer 50

d/dx(loga(x)) = 1/xln(a)

Question 51

What are the steps to solving logarithmic differentiation?

Answer 51

1. Take the natural logarithm of both sides of an equation y = f(x) and use the Laws of logarithms to simplify 2. Differentiate implicitly with respect to x 3. Solve the resulting equation or y'

Question 52

e = ?

Answer 52

Lim(x->0)(1+x)^1/x

Question 53

what are related rates problems?

Answer 53

In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured)

Question 54

What is an Absolute max and min?

Answer 54

Let (c) be a number in the domain D of a function f. Then f(c) is the..... Absolute Maximum Value of f on D if f(c)>/=f(x) for all x in D Absolute Minimum Value of f on D if f(c)=f(x) for all x in D The max and min values are called extreme values of f

Question 55

What is a Local max and min?

Answer 55

The number f(c) is a.... Local Maximum Value of f if f(c)>/=f(x) when (x) is near (c) Local Minimum Value of f if f(c)=f(x) when (x) is near (c)

Question 56

What is the extreme value theorem?

Answer 56

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers (c) and (d) in [a,b] The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. We start by looking for local extreme values

Question 57

What is Fermat's Theorem?

Answer 57

If f has a local maximum or minimum at (c), and is f'(c) exists, then f'(c) = 0

Question 58

What is a Critical Number?

Answer 58

A critical number of a function f is a number (c) in the domain of f such that either f'(c) = 0 or f'(c) = DNE

Question 59

What is closed interval method?

Answer 59

Is a Method used to find the absolute maximum and minimum values of a continuous function f on a closed interval 􏰹[a, b] 1. 􏰺Find the values of f at the critical numbers of f in 􏰨(a, b) 2. 􏰩Find the values of f at the endpoints of the interval 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value

Question 60

What is the increasing and decreasing test (I/D Test)?

Answer 60

You must use the first derivative | a) If f'(x) > 0 on an interval, then f is increasing on an interval (b) If f'(x

Question 61

What is Rolle's theorem?

Answer 61

Let f be a function that satisfies these 3 hypotheses: 1. f is continuous on a closed interval [a,b] 2. f is differentiable on the open interval (a,b) 3. f(a) = f(b) Then there is a number c in 􏰨(a, b)􏰩 such that f'(􏰐􏰨c) = 􏱀 0. In each case it appears that there is at least one point (􏰨c, f(􏰨c))􏰩􏰩 on the graph where the tangent is horizontal and therefore f'(􏰐􏰨c)􏰩 =􏱀 0. Thus Rolle’s Theorem is plausible.

Question 62

What is the Mean Value Theorem?

Answer 62

It is an example of an existence theorem, it guarantees that there exists a number with a certain property, but it doesn't tell us how to find the number. 1. If f is continuous on a closed interval [a,b] 2. If f is differentiable on the open interval (a,b) Then there is a number (c) in (a,b) such that... f'(c) = f(b) - f(a) / b - a or f(b) - f(a) = f'(c)(b - a)

Question 63

What is the first derivative test?

Answer 63

Suppose that (c) is a critical number of a continuous function f. 1. If f' changes from positive to negative at (c), then f has a local maximum at (c) 2. If f' changes from negative to positive at (c), then f has a local minimum at (c) 3. If f' does not change at (c), then f has no local maximum or minimum at (c) The First Derivative Test is a consequence of the I􏰫/D Test

Question 64

What does concave upward mean?

Answer 64

If a graph of f lies above all of its tangents on an interval I, then it is called concave upward. (when the graph lies above the tangent lines, it is concave upward)

Question 65

What does concave downward mean?

Answer 65

If a graph of f lies below all of its tangents on an interval I, then it is called concave downward.

Question 66

What is a concavity test?

Answer 66

You must use the second derivative of a function 1. If f''(x) > 0 for all x in I, then the graph of f is concave upward on I 2. If f''(x)

Question 67

What is a point of inflection?

Answer 67

A point P on a curve y = f(x) is called a Point of Inflection if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P

Question 68

What is the second derivative test?

Answer 68

Is the following test for maximum and minimum values. It is a consequence of the Concavity Test 1. If f'(c) = 0 and f''(c) > 0, then f has a local minimum at (c) 2. If f'(c) = 0 and f''(c)

Question 69

what is an indeterminate?

Answer 69

0/0 or ∞/∞

Question 70

What is L'Hospital's rule?

Answer 70

L’Hospital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. It is especially important to verify the conditions regarding the limits of f and t before using l’Hospital’s Rule. Suppose f and g are differentiable and g'(x) ≠ 0 on an open Interval I, that contains (a) (except possibly at (a)) suppose that... Lim(x->a)f(x) = 0 or +/-∞ and Lim(x->a)g(x) = 0 or +/-∞ Then... Lim(x->a) f(x)/g(x) = Lim(x->a) f'(x)/g'(x)

Question 71

What are the guidelines for Curve sketching?

Answer 71

1. Domain:Find the domain of the function: the set of values of (x) for which f(x) is defined 2. Intercepts: Find the x and y intercepts of the function 3. Find the symmetry of a function about the y-axis f(-x) = f(x) and about the origin f(-x) = -f(x) 4. Find the asymptotes of a the function, whether they are Horizontal (Lim(x->+/-∞) = L, Vertical (Lim(x->a) = -/+∞) or Slant Asymptotes 5. Intervals of Increase/ Decrease: you can find this by using the first derivative test 6. Local Max and Min values: You can find this by using the first or the second derivative test 7. Concavity and Point of Inflection: This can be found by using the concavity test 8. Sketch the Curve

Question 72

What are the steps involved in solving optimization problems?

Answer 72

1. Understand the Problem: The first step is to read the problem carefully until it is clearly understood. Ask yourself: What is the unknown? What are the given quantities? What are the given conditions? 2. Draw a Diagram: In most problems it is useful to draw a diagram and identify the given and required quantities 3. Introduce Notation: Assign a symbol to the quantity that is to be MAXIMIZED or MINIMIZED (ie. Q) also select symbols(a,b,c,d,e,f...) for other unknown quantities and label the diagram with these symbols 4. Express Q in terms of some other symbols from step 3 5. If Q has been expressed as a function of more than one variable, use the given information to find the relationships among these variable (in the form of equations) Then use these equations to eliminate all but one variable (x), say Q = f(x). Write the domain of this function 6. Find the absolute max and minimum value of f. If the domain of f is a closed Interval, then the closed interval method can be used