Chapter 2 Flashcards
Define a limit?
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)
Define one-sided limits?
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
Define Infinite Limits?
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)
what is a vertical asymptote?
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^+) = -∞
what are the limit laws?
- 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)
- 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) - 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) - 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) - 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
what is the direct substitution property?
If f is a polynomial or a rational and (a) is in the domain of f, then
Lim(x-a)f(x) = f(a)
what is the theorem that applies to limits?
If f(x)
what is the squeeze theorem?
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
A number divided by 0 = ?
Undefined
Zero divided by a number = ?
0
The special cubic factors for addition are?
a3 + b3 = (a + b)(a2 – ab + b2)
The special cubic factors for subtraction are?
a3 – b3 = (a – b)(a2 + ab + b2)
Define continuity?
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
what are the 4 things that determine if a function is continuous or not on a specific interval?
- f(a) is defined (that is, (a) is in the domain of f)
- Lim(x-a)f(x) exists
3.Lim(x-a)f(x) = f(a) - 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)
Define discontinuity?
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)
what is the Intermediate Value Theorem?
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)
What are limits to infinity: horizontal asymptotes?
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
What are limits to infinity: horizontal asymptotes negative?
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
what is a horizontal asymptote?
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
To find the horizontal asymptote of a rational function without a square root, you do what?
divide the numerator and the denominator by the highest power of x in the denominator
To find the horizontal asymptote of a rational function that has a square root in the numerator, you do what?
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
what is a tangent line?
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
what is the instantaneous velocity?
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
what are the two derivative definitions?
- 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
- f’(a) = Lim(x->a) f(x) - f(a) / x - a
what are differentiation factors?
f’(x) = y’ = dy/dx = df/dx = d/dxf(x) = Df(x) = Dx f(x)
what determines if a function is differentiable?
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
what determines if a function is non-differentiable?
- 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
- 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
- A third possibility is that the curve has a vertical tangent line when x a
what are higher derivatives? what is a seconds derivative of a function?
This new function f’’ is called the second derivative of f because it is the derivative of the derivative of f
what is the derivative of a constant function?
d/dx (c) = 0
What is the derivative power rule?
d/dx (x^n) = n(x^n-1)
what is the definition of the number e?
Lim(h->0) (e^h -1) / h = 1
what is the derivative of the Natural exponential function (e)?
d/dx (e^x) = e^x
what is a the product rule for differentiation?
fg = f’g + g’f
what is the quotient rule for differentiation?
f/g = f’g - g’f / g^2
Lim(x->0) cosx - 1/x
0
The derivative of sin(x) is?
cos(x)
The derivative of cos(x) is?
-sin(x)
The derivative of tan(x) is ?
sec(x^2)
The derivative of csc(x) is?
-csc(x)cot(x)
The derivative of sec(x) is?
sec(x)tan(x)
The derivative of cot(x) is?
-csc(x^2)
What is the chain rule?
If t is differentiable at x and f is differentiable at tx, 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)
What is implicit differentiation?
This method consists of differentiating both sides of the equation with respect to (x) and then solving the resulting equation for y’
what is the derivative of inverse sin(x) (sinx^-1)?
1/√(1-X^2)
what is the derivative of inverse cos(x) (cosx^-1)?
-1/√(1-X^2)
What is the derivative of inverse tan(x) (tanx^-1)?
1/√(1+X^2)
What is the derivative of inverse cot(x) (cotx^-1)?
-1/√(1+X^2)
What is the derivative of inverse sec(x) (secx^-1)?
1/(x)√(x^2-1)
What is the derivative of inverse csc(x) (cscx^1)?
-1/(x)√(x^2-1)
what is the derivative of a Logarithmic function?
d/dx(loga(x)) = 1/xln(a)
What are the steps to solving logarithmic differentiation?
- Take the natural logarithm of both sides of an equation y = f(x) and use the Laws of logarithms to simplify
- Differentiate implicitly with respect to x
- Solve the resulting equation or y’
e = ?
Lim(x->0)(1+x)^1/x
what are related rates problems?
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)
What is an Absolute max and min?
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
What is a Local max and min?
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)
What is the extreme value theorem?
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
What is Fermat’s Theorem?
If f has a local maximum or minimum at (c), and is f’(c) exists, then f’(c) = 0
What is a Critical Number?
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
What is closed interval method?
Is a Method used to find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]
- Find the values of f at the critical numbers of f in (a, b)
- Find the values of f at the endpoints of the interval
- 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
What is the increasing and decreasing test (I/D Test)?
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
What is Rolle’s theorem?
Let f be a function that satisfies these 3 hypotheses:
- f is continuous on a closed interval [a,b]
- f is differentiable on the open interval (a,b)
- 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.
What is the Mean Value Theorem?
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.
- If f is continuous on a closed interval [a,b]
- 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)
What is the first derivative test?
Suppose that (c) is a critical number of a continuous function f.
- If f’ changes from positive to negative at (c), then f has a local maximum at (c)
- If f’ changes from negative to positive at (c), then f has a local minimum at (c)
- 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
What does concave upward mean?
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)
What does concave downward mean?
If a graph of f lies below all of its tangents on an interval I, then it is called concave downward.
What is a concavity test?
You must use the second derivative of a function
- If f’‘(x) > 0 for all x in I, then the graph of f is concave upward on I
- If f’‘(x)
What is a point of inflection?
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
What is the second derivative test?
Is the following test for maximum and minimum values. It is a consequence of the Concavity Test
- If f’(c) = 0 and f’‘(c) > 0, then f has a local minimum at (c)
- If f’(c) = 0 and f’‘(c)
what is an indeterminate?
0/0 or ∞/∞
What is L’Hospital’s rule?
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)
What are the guidelines for Curve sketching?
- Domain:Find the domain of the function: the set of values of (x) for which f(x) is defined
- Intercepts: Find the x and y intercepts of the function
- Find the symmetry of a function about the y-axis f(-x) = f(x) and about the origin f(-x) = -f(x)
- Find the asymptotes of a the function, whether they are Horizontal (Lim(x->+/-∞) = L, Vertical (Lim(x->a) = -/+∞) or Slant Asymptotes
- Intervals of Increase/ Decrease: you can find this by using the first derivative test
- Local Max and Min values: You can find this by using the first or the second derivative test
- Concavity and Point of Inflection: This can be found by using the concavity test
- Sketch the Curve
What are the steps involved in solving optimization problems?
- 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?
- Draw a Diagram: In most problems it is useful to draw a diagram and identify the given and required quantities
- 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
- Express Q in terms of some other symbols from step 3
- 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
- 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
