Prove that f(x) = x 2 takes on the value 0.5 in the interval [0, 1].

Q: Show that the function f(x) = exp(−|x|) is not differentiable at x = 0.

A: We have given function fx=e-x As we know the modulus function is not differentiable at x=0.…

Q: Verify the Rolle's Theorem for the function (x2-1)(x-2) on [-1,2] find the point in the interval…

A: Given function is : fx=x2-1x-2 To verify the Rolle's Theorem and find the point in the interval…

Q: Find the antiderivative F of f(x)=4-3(1+x^2)^-1 that satisfies F(1)=-2.

A:

Q: Examine if f(x) = ln(x2+2) is uniform continuous on the interval (-∞,∞)

A:

Q: Consider the functions f(x) = cos(x) and g(x) = x--1 over the interval

A: Definite integration can be used to find the volume of a solid. A small volume element is integrated…

Q: : Apply Rolle's theorem for the function f(z) = 7x² – 28z+ 21 over the interval [1,3]. er.

A: The given function is, f(x)=7x2-28x+21 over the interval [1,3]. Observe that, the given function…

Q: Verify Rolle's theorem for f(x) = x* – 4x+8 over the interval [0, 2].

A:

Q: 2. Given the function: f(x) 3 x -3 x≤2 2 x² + 1 x > 2 Is f(x) continuous and differentiable at x=2?

A: Here is the solution:

Q: Is the function a. EVEN h ODD f(x) 1, -π<x<0 1, 0≤x<T

A:

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: Given:          F(x)=∫−2x(t2−2t)dt

Q: Show analytically that the function f is not differentiable at x = x² x1

A:

Q: Prove that for all a, b = R, the function f(x) = 3x² - 4x³ + 6x² + ax + b has exactly one point…

A:

Q: Q1: If the function f(x) define as f: R→ R such that f(x) = (x+1, x>2 x²-1, x≤2 Show that the…

A: This is the question to check the differentiability of the function.

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = 1 + t3 dt

A: To find F'x Fx=∫0xt21+t3dt

Q: Find value a  such that f(x)={x^2+2x-1 if x not equal to 1…

A: given,f(x)=x2+2x-1  if  x≠1a                if x=1left hand limit at x=1limx→1- f(x)=limx→1-…

Q: What is the minimum f(x) value of f(x) = sin x on the interval [0, 2]? %3!

A: Derivative Test: find values of x for which f'(x)=0 Then find f''(x) and put values pf x from above…

Q: Suppose that f is a function on the interval [1, 3] such that –1 < f(x) < 1 for all x and f(x) dx…

A: It is given that ∫13fxdx=0 and -1≤fx≤1.

Q: Show that the functions f (x) = I x I and g(x) = x 1/3 are not differentiable at x = 0

A: Derivative of x at x=0:   Plotting: x,  xx

Q: Use the Intermediate Value Theorem to show that f(x) = cos^−1 (x) − e^x has a zero in the interval…

A: The Intermediate Value Theorem: If f is a function which is continuous at every point of the…

Q: Let 'f be a continuously differentiable real valued function defined bx + a by f(x) ={ 5x x<1 x21…

A:

Q: Determine the domain of the function and prove that it is continuous on its domain f(x) = e - x 2

A: Consider the given function. fx=e-x2 The function is define for every real value of x. Hence,…

Q: Is the function f(x)= x+|x|-1 differentiable at x=0.

A: Consider the given function:

Q: 2) Consider the piecewise function: |x – 1| x² – 1' x 1 For what real number A is the function…

A: A function f(x) is continuous at x=a if the left-hand limit and the right-hand limit of the function…

Q: Differentiate f(x) = f'(x) = 2 ax with respect to x. Assume that a and k are positive constants. k+x

A: To differentiate fx=ax2k+x with respect to x. Assume that a and k are positive constants.

Q: 3. Consider the function Prove that f'(0) = 0. & f(x) = if r EQ if x & Q.

A: Introduction: A differentiable characteristic of 1 actual variable is one whose spinoff takes place…

Q: Show that f(x) = x – cos(x) has only 1 real zero on the interval [0, .

A: given fx=x-cosxx∈0,π2

Q: 6. Use the definition, determine whether the function g(x) = x = 2. Justify the conclusion. x² -…

A: The solution is given in the next step.

Q: Let f be the function defined by f (x) = x cos x – sin x. What is the absolute maximum value of f on…

A: The given function is: f(x)=x cos(x)-sin(x) ;                   x∈-π2, 2π

Q: Let 'f be a continuously differentiable real valued function defined bx+a x<1 by f(x) = 5x² Then the…

A:

Q: Prove that f (x) = x2 takes on the value 0.5 in the interval [0, 1].

A: Here given

Q: Find the best linear approximation to f on  the interval [-1,1].                        f(x) = x2

A:

Q: Show that f(x)= |x| is not differentiable at x = 0.

A:

Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function -5x f(1) = | V1+ tª…

A: To differentiate the function by using first fundamental theorem of calculus f(x)=integration (x to…

Q: F(x) = (3t +2) dt +2

A:

Q: The function f(x, y) -{¹. - x² - y², 0, if if x² + y² 1 , is continuous

A: Given function is fx,y=1−x2−y2, if x2+y2≤10,             if x2+y2&gt;1. We have to check whether the…

Q: Prove that the function f(x) = cos x is strictly decreasing on the interval [0,1]. %3!

A: 5) Given that, fx=cosx

Q: Show that the function f(x) = 1 – I - x² is continuous on the interval [-1, 1].

A:

Q: Use the intermediate value theorem to show that f(x) = e −x 2 − x has at least one root in the…

A:

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: It is given that,Fx=∫-2xt2-2tdt We have to use the Second Fundamental Theorem of Calculus to find…

Q: Show that f(x) := x/3, x e R, is not differentiable at x 0.

A:

Q: The function f (x) ,x € [-27, 2n] is continuous on: 1-cos x

A:

Q: Find the values of a and b such that f is differentiable at x = 3

A: Given,            The function is fx=ax+bx≤32x1-1x&gt;3To find : Find the values of a and b such…

Q: 34. Show that f(x) = Jã is not differentiable at x = 0.

A:

Q: B) Show that f(x) sin for x * 0. for x= 0. is not continuous.

A:

Q: (b) FIND the value of c, so that f(x) = {cx – 1, x< 0 lx² – 1. x 2 0 is continuous everywhere.

A:

Q: Let f : (0,∞) → (0,∞) be given by f(x) = ln(2 + x). Prove that f has a fixed point located between 1…

A: The given problem is to prove that the function f(x) has a fixed point in between 1 and 1.2 Given…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.