Slide 1
... A function P is called a polynomial if P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 where n is a nonnegative integer and the numbers a0, a1, a2, …, an are constants called the coefficients of the polynomial. ...
... A function P is called a polynomial if P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 where n is a nonnegative integer and the numbers a0, a1, a2, …, an are constants called the coefficients of the polynomial. ...
Microsoft Word Format
... Limit: if f(x) becomes arbitrarily close to L as x approaches C from either side, then the limit of f(x) as x approaches C is L lim f(x) = L x Limits that fail to exist: 1. f(x) approaches a different number from the right side of C than from the left side of C 2. f(x) increases or decreases witho ...
... Limit: if f(x) becomes arbitrarily close to L as x approaches C from either side, then the limit of f(x) as x approaches C is L lim f(x) = L x Limits that fail to exist: 1. f(x) approaches a different number from the right side of C than from the left side of C 2. f(x) increases or decreases witho ...
Calc2_RV1
... sums, the definite integral of over [a,b], anti-derivatives of f over [a,b], Fundamental theorem of calculus parts (a) & (b), the average value of f over [a,b]; velocity, speed & acceleration of a particle, the net displacement & the total distance travelled by a particle in a straight line, the nat ...
... sums, the definite integral of over [a,b], anti-derivatives of f over [a,b], Fundamental theorem of calculus parts (a) & (b), the average value of f over [a,b]; velocity, speed & acceleration of a particle, the net displacement & the total distance travelled by a particle in a straight line, the nat ...
The Evaluation Theorem
... Then hand in the solution to your problem on December 3 (the day of the next exam). If I can’t do the problem that you hand in (on November 29) and you show me how to do it (in what you hand in on December 3), then I will replace your two lowest problem scores on the next exam with 10s. If I can do ...
... Then hand in the solution to your problem on December 3 (the day of the next exam). If I can’t do the problem that you hand in (on November 29) and you show me how to do it (in what you hand in on December 3), then I will replace your two lowest problem scores on the next exam with 10s. If I can do ...
Concise
... points of this function. (d) Find the intervals where the graph of this function is concave up and concave down. (e) Sketch the graph of this function. Consider the function y= ...
... points of this function. (d) Find the intervals where the graph of this function is concave up and concave down. (e) Sketch the graph of this function. Consider the function y= ...
Microsoft Word Viewer
... A function is called a piecewise function if it has a different algebraic expression for different parts of its domain. A domain is a collection of numbers on which the function is defined. Piecewise functions are defined in “pieces” because the function behaves differently on some intervals from th ...
... A function is called a piecewise function if it has a different algebraic expression for different parts of its domain. A domain is a collection of numbers on which the function is defined. Piecewise functions are defined in “pieces” because the function behaves differently on some intervals from th ...
File
... 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. This has two important specializations: If a continuous function has values of opposite sign insid ...
... 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. This has two important specializations: If a continuous function has values of opposite sign insid ...
Calculus AB Educational Learning Objectives Science Academy
... Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval Understand and apply the Mean Value Theorem including Rolle’s Theorem Determine intervals on which a function is increas ...
... Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval Understand and apply the Mean Value Theorem including Rolle’s Theorem Determine intervals on which a function is increas ...
Lecture 18: More continuity Let us begin with some examples
... is not uniformly continuous because there is no δ > 0 such that for all x, y ∈ E with |x − y| < δ we have |f (x) − f (y)| < 1. If there were, we could just choose some y ∈ E with |y − x0 | < δ/2 and then deduce that all points z within distance δ of y have f (z) ≤ f (y) + 1. But this is impossible. ...
... is not uniformly continuous because there is no δ > 0 such that for all x, y ∈ E with |x − y| < δ we have |f (x) − f (y)| < 1. If there were, we could just choose some y ∈ E with |y − x0 | < δ/2 and then deduce that all points z within distance δ of y have f (z) ≤ f (y) + 1. But this is impossible. ...
Lecture10
... a) Recall that a random variable X is simply a function from a sample space S into the real numbers. The random variable is discrete is the range of X is finite or countably infinite. This refers to the number of values X can take on, not the size of the values. The random variable is continuous if ...
... a) Recall that a random variable X is simply a function from a sample space S into the real numbers. The random variable is discrete is the range of X is finite or countably infinite. This refers to the number of values X can take on, not the size of the values. The random variable is continuous if ...
SOLUTIONS TO PROBLEM SET 4 1. Without loss of generality
... To show the Feller property take an arbitrary f ∈ C0 (R+ ), i.e. a continuous function on R+ which vanishes at ∞. Then, g : R 7→ R defined by g(x) = f (|x|) belongs to C0 (R). Note that Qt f (x) = Pt g(x), x ≥ 0. Thus, the desired Feller property follows from the Feller property of Brownian motion. ...
... To show the Feller property take an arbitrary f ∈ C0 (R+ ), i.e. a continuous function on R+ which vanishes at ∞. Then, g : R 7→ R defined by g(x) = f (|x|) belongs to C0 (R). Note that Qt f (x) = Pt g(x), x ≥ 0. Thus, the desired Feller property follows from the Feller property of Brownian motion. ...
- Deer Creek High School
... allows students to visualize abstract solutions and understand what they have really found. One of the first topics that we discuss with a graphing calculator is how to properly utilize a calculator. Another activity that students utilize their graphing calculators for is finding the equation of a t ...
... allows students to visualize abstract solutions and understand what they have really found. One of the first topics that we discuss with a graphing calculator is how to properly utilize a calculator. Another activity that students utilize their graphing calculators for is finding the equation of a t ...
1 Introduction and Definitions 2 Example: The Area of a Circle
... Notation 1 If I slip up, it’s likely that I’ll denote x1 ; x2 ; ::: as fxn gn=1 ; or, even shorter, as fxn g. Keep in mind that this is just notation, so you shouldn’t be scared of it. However, I’ll try to avoid building up an excessive amount of notation since that can get confusing. This seems lik ...
... Notation 1 If I slip up, it’s likely that I’ll denote x1 ; x2 ; ::: as fxn gn=1 ; or, even shorter, as fxn g. Keep in mind that this is just notation, so you shouldn’t be scared of it. However, I’ll try to avoid building up an excessive amount of notation since that can get confusing. This seems lik ...
FABER FUNCTIONS 1. Introduction 1 Despite the fact that “most
... encounter are quite nice. It was not until Weierstrass presented his construction of such a function in 1872 that it was widely known that this was possible. Before Weierstrass, there had been a couple of other examples, but they were not published until after Weierstrass told the world about his fu ...
... encounter are quite nice. It was not until Weierstrass presented his construction of such a function in 1872 that it was widely known that this was possible. Before Weierstrass, there had been a couple of other examples, but they were not published until after Weierstrass told the world about his fu ...
06.01-text.pdf
... 2. When we write cos x dx “ sin x ` C, the content of this mathematical statement can be phrased in terms of antiderivatives (as in Question 1). But it can also be phrased in terms of derivatives: ”Functions of the form sin x ` C have, as their derivative, the function cos x.” When viewed this way, ...
... 2. When we write cos x dx “ sin x ` C, the content of this mathematical statement can be phrased in terms of antiderivatives (as in Question 1). But it can also be phrased in terms of derivatives: ”Functions of the form sin x ` C have, as their derivative, the function cos x.” When viewed this way, ...
The Fundamental Theorem of Calculus
... Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the xaxis, and the vertical lines x = 0 and x=2 ...
... Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the xaxis, and the vertical lines x = 0 and x=2 ...
hw1 due - EOU Physics
... 13 + 842 = 852. Determine the next Pythagorean Triplet which starts with 142 + ... (Hint: if you can't figure it out, the answer is in the tutorial, read it!) ...
... 13 + 842 = 852. Determine the next Pythagorean Triplet which starts with 142 + ... (Hint: if you can't figure it out, the answer is in the tutorial, read it!) ...
Day 1
... 2. Find the domain and range of the following functions. (a) f : [r, s, t, u] → [A, B, C, D, E] where f (r) = A, f (s) = B, f (t) = B, and f (u) = E (b) g(t) = t4 (c) f (x) = −x 3. Determine whether the equation defines y as a function of x. (a) x = y 3 (b) x2 + y = 9 4. Sketch f (x) = x2 − 4. Deter ...
... 2. Find the domain and range of the following functions. (a) f : [r, s, t, u] → [A, B, C, D, E] where f (r) = A, f (s) = B, f (t) = B, and f (u) = E (b) g(t) = t4 (c) f (x) = −x 3. Determine whether the equation defines y as a function of x. (a) x = y 3 (b) x2 + y = 9 4. Sketch f (x) = x2 − 4. Deter ...
Lecture Notes for Section 6.1
... Section 6.1: Review of Integration Formulas and Techniques Big idea: With some creative algebra, you can do a lot of “new-looking” integrals by manipulating the integrand to match integral formulas from Calculus 1. Big skill: You should be able to manipulate the integrands of the integrals in this s ...
... Section 6.1: Review of Integration Formulas and Techniques Big idea: With some creative algebra, you can do a lot of “new-looking” integrals by manipulating the integrand to match integral formulas from Calculus 1. Big skill: You should be able to manipulate the integrands of the integrals in this s ...
lesson 29 the first fundamental theorem of calculus
... discovered some members of that family by evaluating the definite integral 2t dt and ...
... discovered some members of that family by evaluating the definite integral 2t dt and ...
PRECALCULUS MA2090 - SUNY Old Westbury
... their graphs. This course is designed primarily for students who wish to take MA2310 Calculus & Analytic Geometry I. COURSE OBJECTIVES: ...
... their graphs. This course is designed primarily for students who wish to take MA2310 Calculus & Analytic Geometry I. COURSE OBJECTIVES: ...
9 Complex-valued Functions
... Ascoli-Arzela Theorem It is known (see Theorem 9.1) that any bounded sequence of real (or complex) numbers has a convergent subsequence. In the theory of functions, one may ask a similar question: For a sequence of bounded continuous functions, is there any uniformly convergent subsequence (so that ...
... Ascoli-Arzela Theorem It is known (see Theorem 9.1) that any bounded sequence of real (or complex) numbers has a convergent subsequence. In the theory of functions, one may ask a similar question: For a sequence of bounded continuous functions, is there any uniformly convergent subsequence (so that ...
Lebesgue integration
In mathematics, the integral of a non-negative function can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.Mathematicians had long understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, we might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job.The Lebesgue integral plays an important role in the branch of mathematics called real analysis, and in many other mathematical sciences fields. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability.The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue—or the specific case of integration of a function defined on a sub-domain of the real line with respect to Lebesgue measure.