1. Consider the function f defined on [0, ∞), 1 f(x) = {x' sin-, x# 0 x = 0 where r > 0. Determine the range of r in which (a) f is continuous on [0, 0), (b) f is differentiable on [0,∞), (c) f' exists and is differentiable on [0, 0).

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Q: (a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0

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Q: 15.1. Using the results of the preceding exercise, show that the function g, defined on R to R by…

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Q: 1. Consider the function f defined on [0, 0), f (x) = {x* sin- x + 0 x = 0 where r > 0. Determine…

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Q: 3. Consider the function g(x) defined by | 22 sin(1/x) if x > 0, g(x) = –x² if x < 0. (a) Show that…

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Q: (b)Consider the piecewise function g(z) = { (1+z)* – 1, if z >0 sin (4z), if z < 0. Discover whether…

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Q: 7. Let f(2) = f(x, y) = + i , = 0. when z +0 and let f(0) %3D 5xº+y² (a) Show that lim f(z) = f(0) =…

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Q: 1. Consider the function Sa(1 – x) sin r€ (0, 1), πε (0, 1) , - f (x) = %3D (0, x € {0, 1}. (a)…

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Q: 2. a) Let u(x, y) be a continous function on a closed and bounded, simply connected region R. Also…

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Q: 3. Show that x? if x>0 0 if x<0 f(x) = is differentiable at xo =0 but f' is not differentiable at xo…

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Q: Define a function g by the rule | 2x sin(1/x) – cos(1/x) if x #0 and g(x) = if x = 0. You may assume…

A: Giveng(x) = 2xsin1x-cos1x   ; x≠0   0                                    ; x=0∫01g(x)dx =…

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Q: 2. Prove that if f: I R is continuous on I, then |f|: I – R where |F(x) := |f(x)| Vz E I is…

A: f is defined on the set of real numbers. The detailed solution is as follows below:

Q: 2. The following funchion O, x = 0 flad = f(x) |-X, 0<x<l is differentiable on (0,1) and satisfies…

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Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y…

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Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…

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Q: II. Determine if [x+1]. if x < -1 S x + 1 f(x) = 2 tanh (x+¹) if 1<x< 1 1-x In [cosh (x + 1) + sinh…

A: It is solved whether f(x) is continuous or not.

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Q: Consider the function f(x) over the interval 0, 1. Does the extreme value theorem guarantee the…

A: Given function is f(x)=1sin πx4 over the interval 0, 1 and  q(x)=4x+32x2-1 Extreme value theorem: If…

Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that f(x)dx = A, / f(x)dæ = B , &nd…

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Q: Let Assume that f has a continuous derivative on a, b. Show tha p(x) such that |f(r) – p(x)| 0. (b)…

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Q: Exercise 1. Express the limit as a definite integral on the given interval cosxi a) lim E-1 Xị Ax,…

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Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.