Flashcards
Functions and limits review: algebra, inequalities, domain
Flashcards do conjunto (94)
- 1IntegersPositive and negative whole numbers (i.e. -3,-2,-1,0,1,2,3)
- 2Real NumbersSplit into two categories:
Rational: a real number that can be written as a quotient of the form (p/q) for some integers p and q (q CANNOT = 0)*
Irrational: a real number that cannot be written in the form of a rational number - 3Interval NotationCertain connected subsets of the real number line (ex. the set of real numbers greater than 0 or the set of real numbers between 0 and 2); these subsets are called intervals*
[0,2]= a closed interval, includes endpoints
(0,2)= open interval, doesn't include endpoints
(-inf,inf)= the entire real number line - 4Set NotationHow we describe compact sets.
Ex: S={x E category/test determining whether x is in the set}
an object of z is an element of a set S if it is contained in S, that is, if it passes the test written in the second half of the proceeding notation
Ex: S={X E R/ X^2>4}; -5 and 3 pass the test, 1 does not, so 1 is not in S.
OR we could write it as {X E R/ X>2 or X<-2} OR (-INF, -2) AND (2,INF) - 5UNIONThe set of elements that are in either A or B (OR) (U)
- 6INTERSECTIONThe set of elements that are in both A and B
- 7Absolute ValueGives us a way to measure the relative positions of numbers on a number line
A.V. makes the number a positive
AV behaves well with products and quotients but not sums. - 8Equationtwo math expressions connected by an = sign
May be true for some values of the variables involved and false for others
Ex: x^2-1=0 is not true for all values of x. In fact it is only true for x=+/- 1 - 9Solution SetThe set of values for which an equation is true; so the solution set of x^2-1=0 is [-1,1].
If an equation involves 2 variables than its solution set consists of ordered pairs (x,y) - 10FactoringCan solve equations by writing the expression as a product of simpler expressions
Ex: 3x^2-6x: 3x(x-2) - 11Theorem 1.1A product is 0 if and only if one of the factors is 0: AB=0 if A=0 or B=0 or both
- 12Quadratic FormulaFor those harder factoring days.
- 13Other Factoring Formulasa^2-b^2=(a-b)(a+b)
a^3-b^3= (a-b)(a^2+ab+b^2) - 14Algebraic Rules for Fractions(a/b)+(c/d)=(ad+bc/bd)
Ex: (1/2)+(1/4)=(4+2)/8=6/8=3/4
(a/b)(c/d)=(ac)/(bd)
Ex: (1/2)(1/4)=(1)/(8)
A/A=1 as long as A does not =0 (why you can cancel common factors) - 15Inequalities (0.3)two math expressions related by a greater than (>), less than (<) or greater than/less signs
The SOLUTION set of an inequality is the set of all values that make the inequality true= an interval of real numbers (vs. equations solution set of finite points). - 16What happens to inequalities when they are inverted or multiplied by a nonzero constant?Multiplying an inequality by a NEGATIVE number "flips" the inequality: for ex. if a < b then -a>-b
We can add or subtract things from both sides of an inequality without flipping the sign. - 17Can you multiply both sides of an inequality by a quantity who's sign you don't know (i.e. x)?NONONONONONONONO
Since multi. both sides by a neg flips the sign, you wont know if you need to flip the sign or not - 18Using inequalities to determine pos. and neg. productsAB>0 if A and B have the same signs (either both are >0 OR <0)
AB<0 If they both have opposite signs - 19Solving Inequalities that involve products or quotientsCOME BACK
- 20When can a polynomial or quotient of a polynomial change signs?A polynomial can change sign only at values where it is = to 0. A quotient of a polynomial can only change sign at values where its numerator or denominator is = to 0
Sign Chart*
- 21Signs and Number LinesWhen the sign is <, it seems the area BETWEEN two points hold solutions and when the sign is > , everywhere but the middle is a solution
- 22FunctionsFunctions describe a certain kind of rule, relationship, or map from the elements of one set to the elements of another set.
A function f from set A to set B is an assignment f that associates to each element c of the domain set A exactly one element f(x) of the codomain or target.*** - 23Independent Variable XX is the input of a function; f sends each input x to one and ONLY one output*
- 24What is not a function?*If a rule assigns a real number input to more than one output, then that rule is not a function (ex. y=+/-sqrt(x)); x has two different y values if x=4 (either 2 or -2)******************
- 25DomainAll possible inputs (is there anything we can't plug into a function)
-denominator as =0 can't happen
sqrt(neg #) can't happen - 26RangeAll possible outcomes: Is there anything we can't get out of a function?
- 27Vertical Line TestA function always has exactly one output value for every input in the domain; a VL can only intersect a graph at one point and only then is it a function. (i.e. no same x more than once): a ONE TO ONE FUNCTION
- 28A function but NOT 1 to 1A graph/function that assigns the same output to two distinct inputs, it is a function, but not one to one. IT DOESNT PASS THE HORIZONTAL LINE TEST.
- 29ROOTSThe x-intercept of the functions, where f(x)=0
- 30Y-interceptWhere the functions crosses the y axis and x=0
- 31Average Rate of Change**of a function f on an interval [a,b] is the slope of the line from (a,f(a)) to (b,f(b)) which is given by (f(b)-f(a))/(b-a)***
Need to look at difference between this and insta ROC - 32AsymptotesWhen a function gets closer and closer to either the X or Y axis but doesn't ever touch it
-Functions can cross the HA
-Functions can not cross the VA - 33Linear Functionsf(x)=mx+b; where m and b are any real numbers (ex. f(x)=2x-1)
-domain is all real #'s
-Range is all real #'s if m is not 0, and is {b} if m=0
-Avg. ROC is m on any interval - 34Power Functionf(x)= Ax^k; where A does not = 0, and k is RATIONAL (ex. 3x^3 or 1.7x^ (-1/5)
- 35Polynomialf (x) = a n x ^n + a n-1 x^ n-1 + ... + a 2 x 2 + a 1 x + a 0
where n= a non negative integer and each a is a real number (ex. 3x^5-2x^3 +x-6) - 36Rational Functionf(x)= (p(x))/(q(x)), where p(x) and q(x) are polynomials (ex. f(x)=(3x^2-1)/(x^5+x^3+1)
- 37Absolute Value FunctionIt is the piecewise defined function; doesn't matter which one is equal to? One has to be. It will leave positive qualities untouched but will flip the sign of neg. quantities so no values will be below the x-axis
- 38Functions as Operators*a function (f) takes an input x and operates on it to perform an output (like a machine).
BUT functions are also objects that can be modified - 39Constant Multiple*the function kf defined by (kf)(x) for all x in the domain of f
- 40Sum rulef and g is the fun. f+g defined by (f+g)(x)=f(x)+g(x) for all x in the domains of both f and g
- 41Product rulethe function f . g define by (f.g)(x)=f(x)g(x) for all x in the domains of both f and g
- 42Quotient Rule(f/g) defined by (f/g)(x)=f(x)/g(x) for all x in the domains
- 43Composition Rule *f o g defined by (fog)(x)=f(g(x))
Note* fog is not necessarily the same as gof - 44Transformationsf(x)+C or f(x)-C shifts it up and down by C
f(x+C) or f(x-c) shifts it left and right
Cf(x) stretches it vertically
(c/x)f(x) compresses it vertically
f(Cx) stretches it horizontally
f((c/x)x) compresses it horizontally
-f(x) flips across X-axis
f(-x) flips across y axis - 45Even Functiona function f is an even function if f(-x)=f(x) for all x in the domain of f
- 46Odd FunctionA function f is an odd function if f(-x)= --f(x)for all x in the domain
(ex: X^2 and x^3 respectively) - 47InverseIn mathematics, the inverse of a function
y=f(x)
y=f(x) is a function that, in some fashion, "undoes" the effect of
f
f (see inverse function for a formal and detailed definition). The inverse of
f
f is denoted
f
−
1
f^{-1}. The statements y = f(x) and x = f −1(y) are equivalent.
*only has an inverse if one to one - 48limitsA limit is what the output of a function approaches as we let the input of that function approach some value. Say we let k approach infinity, we are never actually letting k= infinity, we think about what the ... approaches even if it never actually reaches that value. The lim (x-> c) f(x)=L is not always the same as f(c) because we may or may not reach that point.
As x approaches a c (on the x axis) what y value does it get closer to? y is limit
- 49Limit at infinityif the values of function f(x) approach L as x grows without bound, then we say that L is the limit of f(x) as x approaches inf.
- 50Infinite LimitIf the values of f(x) grow without bound as x approaches c then we say that inf is the limit of f(x) as x approaches c
- 51infinite limit at infinityboth values of f(x) and x grow without bound than we say inf is the limit of f(x) as x approaches inf
- 52Limit existsWhen A limit approaches a real number
- 53Limit DNEIf a limit approaches inf or -inf
- 54When X is approaching c...We are considering values a little less than and a little greater than c
- 55Left hand Limit & Right Hand LimitWhen approaching C from the left, we denote the x->c(-) with a negative
When approaching C from the right, we done X->C(+) with a pos. - 56Two sided LimitExists if and only if the R and L limits as x approaches c exists and are equal
The value of the function at (ex.) x=1 happens to be f(1)=3 but it isn't relevant to the limit calculation - 57Vertical AsymptoteVertical asymptotes are straight lines of the equation , toward which a function f(x) approaches infinitesimally closely, but never reaches the line, as f(x) increases without bound. For these values of x, the function is either unbounded or is undefined.
X->c from either the left/right and it = inf/-inf (the limit is infinity)
If f(x) approaches different signs of infinity from the left and the right then the two sided limit DNE - 58Horizontal AsymptoteFor curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound. (C is the inf or -inf).
- 59Punctured IntervalOnly concerned with what happens AROUND point c
- 60Delta Epsilon Proofs**Trying to prove that EVERY value of e has a corresponding value of delta
Note to self: I would rather look at my notes for steps but WILL go back and do exercises from TB after - 61Continuous Functiona function is continuous if its graph has no breaks, jumps, or holes (you can sketch the graph without picking up your pencil)
- 62Continuity at a Function at a pointf is continuous at x=c if the lim as x approaches c is = to f(c), limit and the point where the function is at c are = to the same number
- 63Right ContinuousAt x=c if lim as x approaches c from the right is = f(c)
- 64Left ContinuousAt x=c if lim as x approaches c from the left is = f(c)
- 65Cont. Open and Closed intervals and graphsCont. on (1,3)= shows a graph with two unfilled circles on 1 and 3 (NOT ROOTS)
cont. on [1,3) with a closed dot at 1
cont. on (1,3] with a closed dot at 3
cont. on [1,3] with two closed dots - 66Types of discontinuities when f is discount. at x=cThis means that the lim of f(x) as x approaches c is NOT EQUAL TO THE VALE F(C)
Removable Discont.- the limit as x approaches c exists but it is not equal to f(c); like a hole punch
Jump Discont.- The left and right limits exist but are not equal to each other
Infinite Discont.- if either the right, left limits, or both is infinite (there is a VA at x=c) - 67A func. is cont. on its domain...If it is cont. on every interval in which it is defined. Constant identity and linear functions are cont. everywhere oN THEIR DOMAIN and power functions are continuous on their DOMAIN
- 68Extreme Value TheoremIf f is cont. on a closed interval [a,b], then there exist values M and m in the interval [a,b] s.t. M is the max value of f(x) on [a,b] and m is the min value of f(x) on [a,b]
* Not necessarily the point on the intervals themselves, just whatever is highest/lowest on graph - 69Intermediate Value Theoremif a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. OR The curve is the function y = f(x),
which is continuous on the interval [a, b],
and w is a number between f(a) and f(b),
Then ...
... there must be at least one value c within [a, b] such that f(c) = w
In other words the function y = f(x) at some point must be w = f(c) - 70When can a function change signs?Only at roots and discontinuities
- 71Evaluating LimitsLimits of cont. functions at domain points by evaluation
*Constant Multiple
*Sum Rule
*Difference Rule
*Product Rule
*Quotient Rule
*Composition Rule
Combos of continuous functions are continuous
*We can now prove most functions are cont. on their domain, if x=c is in the domain* we can calculate the limit via evaluation (ONLY IF X=C IS IN THE DOMAIN) - 72Limits at Non-domain pointsOne thing we can do is cancel common factors, this can happen because f(x) and g(x) * fun. after cancellation differ at x=c, but we don't really care what happens at this point, only what happens as we approach it
- 73Indeterminate Limitswhich means they may or may not exist
- 74Limits Whos Denominators Approach 0 from the Right or LeftIf the limit is of the form (1/0+), meaning its a really small pos. #, then the lim= infinity
If the limit is of the form (1/0-), meaning its a really small neg. #, then the lim= -infinity
*Note does this mean there is a VA?
When this occurs, the function is said to have an infinite limit; hence, you write . Note also that the function has a vertical asymptote at x = c if either of the above limits hold true.
Ex: Example 2: Evaluate .
As x approaches 2 from the left, the numerator approaches 5, and the denominator approaches 0 through negative values; hence, the function decreases without bound and . The function has a vertical asymptote at x = 2. - 75Limits Whose Denominators Become Infinite Approach 0If the lim of f(x)/g(x) as x approaches infinity is equal to (1/-inf) OR (1/inf) then the lim is 0w
- 76Limits of Power Functions @ InfinityIf k>0 then lim of x^k as x goes to inf =inf and if k<0 then lim=0
- 77Indeterminate Formsa limit may not exist, depending on the situation. Need to look back at worksheets to remember how to solve, something like multiplying by the smallest number?*
(0/0)
(inf/inf)
0 . inf
inf-inf
0^0
1^inf
inf^0 - 78Squeeze ThmSuppose f, g, and h are functions so that f(x) ≤ g(x) ≤ h(x)
near a, with the exception that this inequality might not hold when x = a. Then lim f(x) ≤ lim g(x) ≤ lim h(x),
x→a x→a x→a
if these three limits exist. In particular, if limx→a f(x) = L = limx→a h(x), then
lim g(x) = L. x→a - 79Sqrt and ^fraction hintsX^1/k=1->sqrt(x)
x^(p/q)=(q->sqrt(x))^p
x^a.x^b=x^a+b
(xy)^a=x^a.y^a
(x^a)^b=x^ab
(x^a)/(x^b)=x^a-b - 80Quadratic Polynomial Factoring Example3x^2-x-2=(3x + ?)(x + ?)= (3x+1)(x-2)
- 81Cubic Poly. Factoring Examplex^3-3x^2+5x-15=(x^3-3x^2)+(5x-15)=x^2(x-3)+5(x-3)=(x^2+5)(x-3)
-split them up
-take out a common factor
-group - 82Integer Root Thmf P(x) is a polynomial with integer coefficients and if is a zero of P(x) ( P() = 0 ), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .
We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial. Here are the steps:
Arrange the polynomial in descending order
Write down all the factors of the constant term. These are all the possible values of p .
Write down all the factors of the leading coefficient. These are all the possible values of q .
Write down all the possible values of . Remember that since factors can be negative, and - must both be included. Simplify each value and cross out any duplicates.
Use synthetic division to determine the values of for which P() = 0 . These are all the rational roots of P(x) .
Example: Find all the rational zeros of P(x) = x 3 -9x + 9 + 2x 4 -19x 2 .
http://www.sparknotes.com/math/algebra2/polynomials/section4.rhtml - 83Steps to Synthetic DivisionArrange the polynomial in descending order
Write down all the factors of the constant term. These are all the possible values of p .
Write down all the factors of the leading coefficient. These are all the possible values of q .
Write down all the possible values of . Remember that since factors can be negative, and - must both be included. Simplify each value and cross out any duplicates.
Use synthetic division to determine the values of for which P() = 0 . These are all the rational roots of P(x) .
Step 1: Set up the synthetic division.
Step 2: Bring down the leading coefficient to the bottom row.
Step 3: Multiply c by the value just written on the bottom row.
Step 4: Add the column created in step 3.
Write the sum in the bottom row:
Remainder=0 than that number in the corner is a root - 84Polynomial Long DivisionEx:Divide x2 – 9x – 10 by x + 1
org function=(what its divided by, so x+1)(on top of box) +/- remainder
*answer is really whats on top of box since you multiply the rest by 0 - 85Integer Power FunctionsEven and odd
AS the exponent gets higher for either even or odd, the function gets closer to touching the x axis but also gets steeper
Odd: Domain is all real #'s graphs pass through (0,0) and (1,1), symmetric with respect to origin (rotate 180), range all real #
Even: symmetric with respect to y axis, can't get neg # out of this fun, (0,inf) is range - 86Roots, Holes, and VA of Rational Functionsf is not defined at the roots of f(x):: f has roots atria points that are roots of p(x) but now q(x); so only when the top=0
f has HOLES at the points that are roots of both p(x) and q(x); same number in both = a hole
to find a VA= set the bottom to 0 - 87Horizontal Asymptote Degree Rules*The three rules that horizontal asymptotes follow are based on the degree of the numerator, n, and the degree of the denominator, m. If n < m (degree on bottom bigger than top), the horizontal asymptote is y = 0. If n = m, the horizontal asymptote is y = a/b. If n > m (degree on top bigger), there is no horizontal asymptote
*As a reminder these are just rules, you to verify the answer by solving CHECK TEST 2 ON HOW - 88Slant & Curve Asymptote*Use polynomial long division so f(x)=(p(x))/(q(x))= ax + ((R(x))/(q(x))
a(x) just = slant?
See test 2*** - 89Slope FunctionsIf a function's graph is smooth, then we can consider the slope at every point in the function.
A tangent line is a straight line that touches a function at only one point. The tangent line represents the instantaneous rate of change of the function at that one point. The slope of the tangent line at a point on the function is equal to the derivative of the function at the same point - 90Insta Rate of ChangeThe instantaneous rate of change tells you how fast y is changing with respect to x at any value of x. It is also called the derivative, and it is the slope of the line tangent to a graph at any point. Analogy of the direction the headlights of a car are pointing at that point.
How is the function changing AT a specific point/mile marker - 91Average Rate of Changethe average rate of change of f(x) on the interval a ≤ x ≤ b is defined as
m = Δy = f (b) − f (a) . Δx b−a
Geometrically, this is the slope of the secant drawn to the graph over the interval [a, b]. - 92Connections between f and f'(x)>For each x, the slope of f(x) is the height of f'(x)
> where f has a HTL, the derivative of f' has a root
Where f is increasing, f' is above the x- axis
where f is decreasing, f' is below the x-axis - 93Approximating the Slope of a Tangent LineThe slope of the tangent line measures the direction of the function, but we can't use slope formula with only one point, so we must use nearby points on the function.
- 94Secant Line*from a to b for a function f is the line that passes through the points (a, f(a)) and (b,f(b)). As we move closer to the point x=c, then we will get secant lines closer to the tangent line we are interested in.
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- FlashCards sobre Sem título, criado por Emily WIlliams em 25-04-2017.
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