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Sub-problem 2. Recall the eztreme value theorem: If g(ax) is continuous on (0, 1], then g attains its marimum. In particular it is BOUNDED. L.e. there is a number M20 such that 1. Find a number M...
s h) for all z c l e Sub-problem 3. Recall monotonicity of integration: If h() S [-1, 1], then This just says that integruls preserve inequalities 1. Explain why this is true graphically 2·Let...
s h) for all z c l e Sub-problem 3. Recall monotonicity of integration: If h() S [-1, 1], then This just says that integruls preserve inequalities 1. Explain why this is true graphically 2·Let g be continuous on [0,1]. Use the previous item, and the fact that to show that 3. Use the first two items to show that if g is bounded, say Ig(r)l s M for z [0, 1], then first two derivatives are continos on is...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prov...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...
Problem 1. (The golden mean] In this problem you will find the exact value of the...
Problem 1. (The golden mean] In this problem you will find the exact value of the number 7, often called the golden mean or the golden ratio (sometimes this terminology is used for 7-1). The golden mean is defined by the following expression: 7= 1+- 1+ - 1 1+ 1+... (a) Consider the iteration Xn+1 = f(xn), where x1 = 1, and 1 f(x) = 1+2 1 1+: Recall the following result. Theorem. (i) If the function g : [a,...
all a,b,c,d 1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on...
all a,b,c,d
1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on C with gradient Vg, we define the normal derivative Dng by and we define the Laplacian, V2g, of g by For this problem, assume D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f...
Problem 3. Prove Theorem 1 as tollows [Assume all conditions of the Theorem are met. In many part...
Problem 3. Prove Theorem 1 as tollows [Assume all conditions of the Theorem are met. In many parts, it will be useful to consider the sign of the right side of the formula-positive or negative- ad to write the appropriate inequality] (a) Since f"(x) exists on [a, brx) is continuous on [a, b) and differentiable on (a,b), soMean Value Thorem applies to f,on this interval. Apply MVTtof"m[x,y], wherc α zcysb. to show that ry)2 f,(x), İ.e. that f, is increasing...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typica...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can'...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in...
Consider the generalized integrator function (2) discussed in class, defined by its proper- ties: | dr...
Consider the generalized integrator function (2) discussed in class, defined by its proper- ties: | dr 8(x) = 1, Ve > 0, | dx 8(x) = 12+ = ſo if r* <0 11 if x* 20' dx 8(2 – c)f(x) = f(c), VER, where dc 8() is understood as a slight abuse of notation and f(x) in the last formula is a suitably well-behaved (at least bounded and continuous - and perhaps even smoother- in a neighborhood of x=c) function...
Find the solution y of the initial value problem 3"(t) = 2 (3(t). y(1) = 0,...
Find the solution y of the initial value problem 3"(t) = 2 (3(t). y(1) = 0, y' (1) = 1. +3 g(t) = M Solve the initial value problem g(t) g” (t) + 50g (+)? = 0, y(0) = 1, y'(0) = 7. g(t) = Σ Use the reduction order method to find a second solution ya to the differential equation ty" + 12ty' +28 y = 0. knowing that the function yı(t) = + 4 is solution to that...
number 1 and 2 pls Problem 1.1. Suppose that f: R → R and that f...
number 1 and 2 pls
Problem 1.1. Suppose that f: R → R and that f is differentiable at z = a. 1. Show that, given an angle 6, we can choose 6(0) > 0 small enough so that for all r such that r - al < (0) we have that the graph of f(r) lies inside of the cone with angle e around the tangent line. 2. Can you find explicit formulas for 6(0) for the function f(x)...
s h) for all z c l e Sub-problem 3. Recall monotonicity of integration: If h() S [-1, 1], then This just says that integruls preserve inequalities 1. Explain why this is true graphically 2·Let g be continuous on [0,1]. Use the previous item, and the fact that to show that 3. Use the first two items to show that if g is bounded, say Ig(r)l s M for z [0, 1], then first two derivatives are continos on is...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...
Problem 1. (The golden mean] In this problem you will find the exact value of the number 7, often called the golden mean or the golden ratio (sometimes this terminology is used for 7-1). The golden mean is defined by the following expression: 7= 1+- 1+ - 1 1+ 1+... (a) Consider the iteration Xn+1 = f(xn), where x1 = 1, and 1 f(x) = 1+2 1 1+: Recall the following result. Theorem. (i) If the function g : [a,...
all a,b,c,d
1. Suppose C is simple closed curve in the plane given by the parametric equation and recall that the outward unit normal vector n to C is given by y(t r'(t) If g is a scalar field on C with gradient Vg, we define the normal derivative Dng by and we define the Laplacian, V2g, of g by For this problem, assume D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f...
Problem 3. Prove Theorem 1 as tollows [Assume all conditions of the Theorem are met. In many parts, it will be useful to consider the sign of the right side of the formula-positive or negative- ad to write the appropriate inequality] (a) Since f"(x) exists on [a, brx) is continuous on [a, b) and differentiable on (a,b), soMean Value Thorem applies to f,on this interval. Apply MVTtof"m[x,y], wherc α zcysb. to show that ry)2 f,(x), İ.e. that f, is increasing...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in...
Consider the generalized integrator function (2) discussed in class, defined by its proper- ties: | dr 8(x) = 1, Ve > 0, | dx 8(x) = 12+ = ſo if r* <0 11 if x* 20' dx 8(2 – c)f(x) = f(c), VER, where dc 8() is understood as a slight abuse of notation and f(x) in the last formula is a suitably well-behaved (at least bounded and continuous - and perhaps even smoother- in a neighborhood of x=c) function...
Find the solution y of the initial value problem 3"(t) = 2 (3(t). y(1) = 0, y' (1) = 1. +3 g(t) = M Solve the initial value problem g(t) g” (t) + 50g (+)? = 0, y(0) = 1, y'(0) = 7. g(t) = Σ Use the reduction order method to find a second solution ya to the differential equation ty" + 12ty' +28 y = 0. knowing that the function yı(t) = + 4 is solution to that...
number 1 and 2 pls
Problem 1.1. Suppose that f: R → R and that f is differentiable at z = a. 1. Show that, given an angle 6, we can choose 6(0) > 0 small enough so that for all r such that r - al < (0) we have that the graph of f(r) lies inside of the cone with angle e around the tangent line. 2. Can you find explicit formulas for 6(0) for the function f(x)...
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