5.1 (page 322-331)
... It is likely that you have studied logarithms in an algebra course. There, without the benefit of calculus, logarithms would have been defined in terms of a base number. For example, common logarithms have a base of 10 and therefore log1010 1. (You will learn more about this in Section 5.5.) The b ...
... It is likely that you have studied logarithms in an algebra course. There, without the benefit of calculus, logarithms would have been defined in terms of a base number. For example, common logarithms have a base of 10 and therefore log1010 1. (You will learn more about this in Section 5.5.) The b ...
3 Lipschitz condition and Lipschitz continuity
... is continuous and if there is a Λ > 0 such that (26) is satisfied for all z, y in Rm and s ∈ R. (Notice again the secondary role played by the time variable.) A fact that we have used in the proof of uniqueness is the following: 3.17 Lemma (Characterization of local Lipschitz-continuity) A continuou ...
... is continuous and if there is a Λ > 0 such that (26) is satisfied for all z, y in Rm and s ∈ R. (Notice again the secondary role played by the time variable.) A fact that we have used in the proof of uniqueness is the following: 3.17 Lemma (Characterization of local Lipschitz-continuity) A continuou ...
Lecture 33: Calculus and Music A music piece is a function The
... Decomposition in overtones: low and high pass filter Every wave form can be written as a sum of sin and cos functions. Our ear does this so called Fourier decomposition automatically. We can here melodies. Here is an example of a decomposition: f (x) = sin(x) + sin(2x)/2 + sin(3x)/3 + sin(4x)/4 + si ...
... Decomposition in overtones: low and high pass filter Every wave form can be written as a sum of sin and cos functions. Our ear does this so called Fourier decomposition automatically. We can here melodies. Here is an example of a decomposition: f (x) = sin(x) + sin(2x)/2 + sin(3x)/3 + sin(4x)/4 + si ...
Lecture 33: Calculus and Music A music piece is a function The
... Decomposition in overtones: low and high pass filter Every wave form can be written as a sum of sin and cos functions. Our ear does this so called Fourier decomposition automatically. We can here melodies. Here is an example of a decomposition: f (x) = sin(x) + sin(2x)/2 + sin(3x)/3 + sin(4x)/4 + si ...
... Decomposition in overtones: low and high pass filter Every wave form can be written as a sum of sin and cos functions. Our ear does this so called Fourier decomposition automatically. We can here melodies. Here is an example of a decomposition: f (x) = sin(x) + sin(2x)/2 + sin(3x)/3 + sin(4x)/4 + si ...
On the building blocks cif mathematical logic
... if;= Trp is that function !f;xy whose value !f;xy coincides with rpyx for all argument values x and y for which rpyx has meaning. We write this definition briefly as ...
... if;= Trp is that function !f;xy whose value !f;xy coincides with rpyx for all argument values x and y for which rpyx has meaning. We write this definition briefly as ...
4. Growth of Functions 4.1. Growth of Functions. Given functions f
... that f is big-O of g expresses the fact that for large enough x, f will be bounded above by some constant multiple of g. Theorem 4.2.1 gives a necessary condition for f to be big-O of g in terms of limits. The two notions aren’t equivalent since there are examples where the definition holds, but the ...
... that f is big-O of g expresses the fact that for large enough x, f will be bounded above by some constant multiple of g. Theorem 4.2.1 gives a necessary condition for f to be big-O of g in terms of limits. The two notions aren’t equivalent since there are examples where the definition holds, but the ...
Section 9.7
... graph of a function is used to obtain a linear approximation of the function near the point of tangency. In this section, we will consider how one might improve on the accuracy of local linear approximations by using higher-order polynomials as approximating functions. We will also investigate t ...
... graph of a function is used to obtain a linear approximation of the function near the point of tangency. In this section, we will consider how one might improve on the accuracy of local linear approximations by using higher-order polynomials as approximating functions. We will also investigate t ...
8.2 Integration by Parts
... Notice that the integrand looks much simpler. At this point we can finish off by using a substitution: u = x3 ...
... Notice that the integrand looks much simpler. At this point we can finish off by using a substitution: u = x3 ...
Calculus I, Fall 2012 - Solutions to Review Problems II
... = − sin x + cos x. So the tangent line is horizontal (i.e., = 0) at all points x such dx dx that cos x = sin x. This means that the point (cos x, sin x) on the unit circle corresponds to an angle of π/4 or 5π/4 = π/4 + π. It follows that x = π/4 + nπ for n = 0, ±1, ±2, . . .. ...
... = − sin x + cos x. So the tangent line is horizontal (i.e., = 0) at all points x such dx dx that cos x = sin x. This means that the point (cos x, sin x) on the unit circle corresponds to an angle of π/4 or 5π/4 = π/4 + π. It follows that x = π/4 + nπ for n = 0, ±1, ±2, . . .. ...
Poisson`s remarkable calculation
... The subject of functional equations, which originated with Cauchy and Abel, has spawned an extensive body of advanced techniques (see, e.g. [A]). These techniques have been used to prove far more general results than those presented here (cf. [Ba], [L] and [M]). The advantage of the present approach ...
... The subject of functional equations, which originated with Cauchy and Abel, has spawned an extensive body of advanced techniques (see, e.g. [A]). These techniques have been used to prove far more general results than those presented here (cf. [Ba], [L] and [M]). The advantage of the present approach ...
MATH 115 ACTIVITY 1:
... Become familiar with the derivative and antiderivative rules for sine and cosine functions: D sin u (cos u )Du D cos u ( sin u )Du ...
... Become familiar with the derivative and antiderivative rules for sine and cosine functions: D sin u (cos u )Du D cos u ( sin u )Du ...
Parent Functions and Transformations Objectives • Identify, graph
... Example 1: Describe the following characteristics of the graph of the parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. a) f(x) = ...
... Example 1: Describe the following characteristics of the graph of the parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. a) f(x) = ...
Module Handbook - Ulster University
... Section A contains ten compulsory questions each worth 4 marks. Section B contains five questions of which the candidates should answer three. ...
... Section A contains ten compulsory questions each worth 4 marks. Section B contains five questions of which the candidates should answer three. ...
Form Properties24
... xz - 1 lim zx•l• x -12xz 2xz lim z x - 1 - , lim z = •x -1 Therefore, the lines x = 1 and x = -1 are vertical asymptotes. This information about limits and asymptotes enables us to draw the preliminary sketch in Figure 8, showing the parts of the curve near the asymptotes. E· f,(x) - 4x (x z - 1 ) - ...
... xz - 1 lim zx•l• x -12xz 2xz lim z x - 1 - , lim z = •x -1 Therefore, the lines x = 1 and x = -1 are vertical asymptotes. This information about limits and asymptotes enables us to draw the preliminary sketch in Figure 8, showing the parts of the curve near the asymptotes. E· f,(x) - 4x (x z - 1 ) - ...
Indefinite Integrals Calculus
... The following applet lets you explore Riemann Sums of any function. You can change the bounds and the number of partitions. Follow the examples given on the page, and then use the applet to explore on your own. Riemann Sums Applet. Note: On this page the author uses Left- and Right- hand sums. These ...
... The following applet lets you explore Riemann Sums of any function. You can change the bounds and the number of partitions. Follow the examples given on the page, and then use the applet to explore on your own. Riemann Sums Applet. Note: On this page the author uses Left- and Right- hand sums. These ...
Sections 1.10-1.11
... (If this is positive, this can be thought of as the speedometer reading at 1pm.) Re Example 1: Definitions • Let t = the time (in hours) elapsed since noon. • Let y = s t , where s is the position function for the car (in miles). ...
... (If this is positive, this can be thought of as the speedometer reading at 1pm.) Re Example 1: Definitions • Let t = the time (in hours) elapsed since noon. • Let y = s t , where s is the position function for the car (in miles). ...
M101 Tut4_SolnD
... Since the planes are parallel, they have the same normal vector. If O is the origin, then it should be clear that the distances l1 and l2 are respectively the scalar projections of OP and OQ onto the normal vector n, where P and Q are any two points on 1 and 2 respectively. Recall that these two ...
... Since the planes are parallel, they have the same normal vector. If O is the origin, then it should be clear that the distances l1 and l2 are respectively the scalar projections of OP and OQ onto the normal vector n, where P and Q are any two points on 1 and 2 respectively. Recall that these two ...
Solution - Harvard Math Department
... The anti-derivative of the function inside the integral is 1/x which does not look good in the limit x → 0. Nope, the integral does not exist. Intuitively, 1/x2 just goes to infinity too fast if x → 0. ...
... The anti-derivative of the function inside the integral is 1/x which does not look good in the limit x → 0. Nope, the integral does not exist. Intuitively, 1/x2 just goes to infinity too fast if x → 0. ...
Week #2:
... Consequently the market is complete. Although the number of possible states is uncountable and there exist only two independent primary assets the market model built from Brownian motions is complete. The value of the derivative at time (t) must be Vt to prevent arbitrage; Vt= BtEt = Bt EQ ( BT1 X ...
... Consequently the market is complete. Although the number of possible states is uncountable and there exist only two independent primary assets the market model built from Brownian motions is complete. The value of the derivative at time (t) must be Vt to prevent arbitrage; Vt= BtEt = Bt EQ ( BT1 X ...
Test #3 Topics
... In Section 2.1, we used the idea of instantaneous velocity to introduce the concept of a limit. In Section 3.1 (and again in Section 3.5) that idea was developed further to introduce the derivative. To compute the distance traveled by an object moving along a straight line at constant velocity (e.g. ...
... In Section 2.1, we used the idea of instantaneous velocity to introduce the concept of a limit. In Section 3.1 (and again in Section 3.5) that idea was developed further to introduce the derivative. To compute the distance traveled by an object moving along a straight line at constant velocity (e.g. ...
Calculus Challenge #7 SOLUTION
... A student was wondering why I took off points for his solution to the following integration problem. ...
... A student was wondering why I took off points for his solution to the following integration problem. ...
Homework 4 Solutions - Math-UMN
... Let > 0 be given. Since f has limit F at x0 and g has limit G at x0 , we know that there exists δf such that if 0 < |x − x0 | < δf and x ∈ D, then |f (x) − F | < /2; and there exists δg such that if 0 < |x − x0 | < δg and x ∈ D, then |g(x) − G| < /2. Take δ = min{δf , δg }. Then for x ∈ D with 0 ...
... Let > 0 be given. Since f has limit F at x0 and g has limit G at x0 , we know that there exists δf such that if 0 < |x − x0 | < δf and x ∈ D, then |f (x) − F | < /2; and there exists δg such that if 0 < |x − x0 | < δg and x ∈ D, then |g(x) − G| < /2. Take δ = min{δf , δg }. Then for x ∈ D with 0 ...
9 - SFU Computing Science
... Models (First Order Semantics) Let be a vocabulary. A model appropriate to is a pair M=(U,) consisting of • the universe of M, a non-empty set U • the interpretation, a function that assigns - to each predicate symbol P a concrete predicate P M on U M - to each function symbol f a concrete fu ...
... Models (First Order Semantics) Let be a vocabulary. A model appropriate to is a pair M=(U,) consisting of • the universe of M, a non-empty set U • the interpretation, a function that assigns - to each predicate symbol P a concrete predicate P M on U M - to each function symbol f a concrete fu ...
AP Calculus AB Hands-On Activity: Rolle`s and Mean Value
... 8. Translate the Mean Value Theorem into a statement about rates of change. At least once on the interval [a,b] the instantaneous rate of change quals the average rate of change for that interval. 9. Rolle's Theorem is a specific case of the MVT, which applies whenever g(a)=g(b). What is the slop ...
... 8. Translate the Mean Value Theorem into a statement about rates of change. At least once on the interval [a,b] the instantaneous rate of change quals the average rate of change for that interval. 9. Rolle's Theorem is a specific case of the MVT, which applies whenever g(a)=g(b). What is the slop ...