spl7.tex Lecture 7. 24.10.2011. Absolute continuity. Theorem. If f ∈ L
... Recall the Fundamental Theorem of Calculus (FTC), that says that differentiation and integration are inverse processes (there are various ways in which this can be made precise). Now that we have a new definition of integration, we need a new version of FTC, as follows. Theorem (Lebesgue’s differentia ...
... Recall the Fundamental Theorem of Calculus (FTC), that says that differentiation and integration are inverse processes (there are various ways in which this can be made precise). Now that we have a new definition of integration, we need a new version of FTC, as follows. Theorem (Lebesgue’s differentia ...
hw1 due - EOU Physics
... Fill in the table below: As before, you may use any source. However, keep in mind that the goal is to actually learn the symbols. (Hint: Many of the answers are at ...
... Fill in the table below: As before, you may use any source. However, keep in mind that the goal is to actually learn the symbols. (Hint: Many of the answers are at ...
Calculus Maximus WS 2.1: Tangent Line Problem
... Remember, parallel lines have the same slope, but different base camps. ...
... Remember, parallel lines have the same slope, but different base camps. ...
Quiz 12 - FAU Math
... to determine the x-coordinates of the local maxima and minima of f , and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). Since f ′ (x) does not fail to exist for all x, the critical points of f are x = −2, x = 0, x = 1 because at those three points f ′ is 0. • ...
... to determine the x-coordinates of the local maxima and minima of f , and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). Since f ′ (x) does not fail to exist for all x, the critical points of f are x = −2, x = 0, x = 1 because at those three points f ′ is 0. • ...
Solutions 1. - UC Davis Mathematics
... • The velocity is zero when t = 1/3 or t = 1. • The acceleration of the particle is d2 s = −4 + 6t, dt2 which is zero if t = 2/3. The location of the particle at that time is s = 2/27. • To answer the question as stated, the particle moves forward from s = 0 to s = 4/27 for 0 ≤ t ≤ 1/3, when its vel ...
... • The velocity is zero when t = 1/3 or t = 1. • The acceleration of the particle is d2 s = −4 + 6t, dt2 which is zero if t = 2/3. The location of the particle at that time is s = 2/27. • To answer the question as stated, the particle moves forward from s = 0 to s = 4/27 for 0 ≤ t ≤ 1/3, when its vel ...
1.5 Function Notation
... previous examples, the independent variable is t, not x. In this context, t is chosen because it represents time. The second is that the function is broken up into two rules: one formula for values of t between 0 and 20 inclusive, and another for values of t greater than 20. To find h(10), we first ...
... previous examples, the independent variable is t, not x. In this context, t is chosen because it represents time. The second is that the function is broken up into two rules: one formula for values of t between 0 and 20 inclusive, and another for values of t greater than 20. To find h(10), we first ...
Mathematical Analysis Worksheet 8
... Recall the Intermediate Value Theorem: Let f : [a, b] → R be continuous and k ∈ (f (a), f (b)) or k ∈ (f (b), f (a). Then there exists c ∈ (a, b) such that f (c) = k. Notes: 1. This result is very useful to prove existence of solutions to equations. 2. It does not tell us where a solution lies in th ...
... Recall the Intermediate Value Theorem: Let f : [a, b] → R be continuous and k ∈ (f (a), f (b)) or k ∈ (f (b), f (a). Then there exists c ∈ (a, b) such that f (c) = k. Notes: 1. This result is very useful to prove existence of solutions to equations. 2. It does not tell us where a solution lies in th ...
Pre calculus Topics
... next year? What should I do if we run into so many snow days again next year/ what should I do if there are fewer snow days? For example I was thinking of running Saturday Prep Session(s) of BC calculus students during the first semester. ...
... next year? What should I do if we run into so many snow days again next year/ what should I do if there are fewer snow days? For example I was thinking of running Saturday Prep Session(s) of BC calculus students during the first semester. ...
Computing Derivatives and Integrals
... Here are a bunch of (indefinite) integrals you should probably know, or be able to compute. Notice that in almost every case, this is just the reverse of the rule you used for derivatives! Remark 1.2. Note that there’s always a +c added to the end of every indefinite integral. This is because an ind ...
... Here are a bunch of (indefinite) integrals you should probably know, or be able to compute. Notice that in almost every case, this is just the reverse of the rule you used for derivatives! Remark 1.2. Note that there’s always a +c added to the end of every indefinite integral. This is because an ind ...
Calculus I - Chabot College
... apply implicit differentiation to solve related rate problems; apply the Mean Value Theorem; demonstrate an understanding of the definite integral as the limit of a Riemann sum; demonstrate an understanding of the Fundamental Theorem of Integral Calculus; demonstrate an understanding of differential ...
... apply implicit differentiation to solve related rate problems; apply the Mean Value Theorem; demonstrate an understanding of the definite integral as the limit of a Riemann sum; demonstrate an understanding of the Fundamental Theorem of Integral Calculus; demonstrate an understanding of differential ...
exponential-log-functions-test-paper-one-2016
... MM UNIT 2 Exponential and Logarithmic Functions Test 2016 Paper 1 Technology Free ...
... MM UNIT 2 Exponential and Logarithmic Functions Test 2016 Paper 1 Technology Free ...
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 18: More continuity Let us begin with some examples
... Here there is not even a limit as x → 0. This is because we can find a sequences (xn ) converging to 0 such that (f (xn )) does not have a limit. Take xn = 2/(nπ) . Derivatives: Ch. 5 Continuous functions are nicer than most functions. However we have seen that they can still be rather weird (recall ...
... Here there is not even a limit as x → 0. This is because we can find a sequences (xn ) converging to 0 such that (f (xn )) does not have a limit. Take xn = 2/(nπ) . Derivatives: Ch. 5 Continuous functions are nicer than most functions. However we have seen that they can still be rather weird (recall ...
Chapter 3 Supplementary Problems
... bacteria doubles in three hours, in how many hours will the number of bacteria triple? Answer exactly. 8. On [0,3], what is the maximum acceleration attainted by the particle whose velocity is given by v(t) = t3 – 3t2 + 12t + 4? ...
... bacteria doubles in three hours, in how many hours will the number of bacteria triple? Answer exactly. 8. On [0,3], what is the maximum acceleration attainted by the particle whose velocity is given by v(t) = t3 – 3t2 + 12t + 4? ...
Calculus II - Chabot College
... perform basic vector algebra in R2 and R3 and interpret the results geometrically; find equations of lines and planes in R3; construct polynomial approximations (Taylor polynomials) for various functions and estimate their accuracy using an appropriate form of the remainder term in Taylor’s formula; ...
... perform basic vector algebra in R2 and R3 and interpret the results geometrically; find equations of lines and planes in R3; construct polynomial approximations (Taylor polynomials) for various functions and estimate their accuracy using an appropriate form of the remainder term in Taylor’s formula; ...
Concise
... Find two positive numbers whose product is 36 and whose sum is a minimum. A drug company is designing an open-top rectangular box with a square base, that will hold 32000 cubic centimeters. Determine the dimensions x and y that will yield the minimum surface area. An open-top box is to be constructe ...
... Find two positive numbers whose product is 36 and whose sum is a minimum. A drug company is designing an open-top rectangular box with a square base, that will hold 32000 cubic centimeters. Determine the dimensions x and y that will yield the minimum surface area. An open-top box is to be constructe ...
math318hw1problems.pdf
... Let f : Y → Z be continuous. For every open V ⊂ Z and for every g ∈ C 0 (V ) we obtain f ∗ g ∈ C 0 (f −1 V ) giiven by (f ∗ g)x = g(f (x)) for all x ∈ f −1 V . Definition 3.2. Let Y be a topological space. A subpresheaf (or a presheaf of subsets) R of CY0 consists of (1) the data: a subset R(U ) ⊂ C ...
... Let f : Y → Z be continuous. For every open V ⊂ Z and for every g ∈ C 0 (V ) we obtain f ∗ g ∈ C 0 (f −1 V ) giiven by (f ∗ g)x = g(f (x)) for all x ∈ f −1 V . Definition 3.2. Let Y be a topological space. A subpresheaf (or a presheaf of subsets) R of CY0 consists of (1) the data: a subset R(U ) ⊂ C ...
Applications of the 2nd Derivative
... If f ′′ ( x ) > 0 then f ′ ( x ) is increasing so the curve is concave up. If f ′′ ( x ) < 0 then f ′ ( x ) is decreasing so the curve is concave down ...
... If f ′′ ( x ) > 0 then f ′ ( x ) is increasing so the curve is concave up. If f ′′ ( x ) < 0 then f ′ ( x ) is decreasing so the curve is concave down ...
Chapter 03 - Mathematical Marketing
... direction in which f is less. On the other hand, if the derivative is positive, as it would be at position 1, we need to move to our left. In more formal terms, in nonlinear optimization we could calculate the next estimate of using the formula ...
... direction in which f is less. On the other hand, if the derivative is positive, as it would be at position 1, we need to move to our left. In more formal terms, in nonlinear optimization we could calculate the next estimate of using the formula ...
FABER FUNCTIONS 1. Introduction 1 Despite the fact that “most
... people are surprised when they hear of this result. In fact, many people are surprised that there even exists a continuous and nowhere differentiable function, as the functions we usually encounter are quite nice. It was not until Weierstrass presented his construction of such a function in 1872 tha ...
... people are surprised when they hear of this result. In fact, many people are surprised that there even exists a continuous and nowhere differentiable function, as the functions we usually encounter are quite nice. It was not until Weierstrass presented his construction of such a function in 1872 tha ...
Mean Value Theorem by Nihir Shah
... the section. It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval. You must be like “what in the world is this?” Allow me to explain in simple terms: ...
... the section. It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval. You must be like “what in the world is this?” Allow me to explain in simple terms: ...
Differentiation - Keele Astrophysics Group
... instantaneous velocity (speed plus direction, which is given by the sign of the slope) of the object at that point and time. Suppose we want to work out how fast the object is travelling at any moment in time. Speed (or velocity, if we care about the sign) is defined as the rate of change of distanc ...
... instantaneous velocity (speed plus direction, which is given by the sign of the slope) of the object at that point and time. Suppose we want to work out how fast the object is travelling at any moment in time. Speed (or velocity, if we care about the sign) is defined as the rate of change of distanc ...
Example 1: Solution: f P
... Use various techniques of differentiations to find the derivatives of various functions. ...
... Use various techniques of differentiations to find the derivatives of various functions. ...
Calculus AB Educational Learning Objectives Science Academy
... PreCalculus Unit The learner will • Solve and graph linear and quadratic inequalities • Define and apply the concept of absolute value • Relate absolute value to distance • Apply the distance formula • Write equations of circles • Find x and y intercepts of functions • Find points of intersection f ...
... PreCalculus Unit The learner will • Solve and graph linear and quadratic inequalities • Define and apply the concept of absolute value • Relate absolute value to distance • Apply the distance formula • Write equations of circles • Find x and y intercepts of functions • Find points of intersection f ...