If     limx→3f(x)=7 ,  which of the following must be true?    I.    f   is continuous at   x=3. II.   f   is differentiable at  x=3. III.     f(3)=7

Q: (c) Suppose that ƒ hits every value between y = 0 and y = 1 on the interval [0, 1]. Must ƒ be…

A: A function is continuous when its graph is a single unbroken curve , that you could draw without…

Q: Consider g(x) = x!/5. Is g(x) differentiable at x=0? Is g(x) continuous at x=0?

A: Given function

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A: see below the answer

Q: If f is continuous on [a, b] and "If(x)| dx = 0, what can you conclude about f ?

A: Solution:Given If f is continuous on a,b and ∫abf(x) dx = 0To conclude about f ?

Q: Let f(x) = { *, if x= 2 ; for the eax if x + 2 function to be continuous at x = 2 the value of a is

A:

Q: In the given equation as follows , determine whether the statement is true or false. If it is false,…

A: f is continuous function on [0,∞) and ∫0∞f(x)dx diverges

Q: Show that the function f(x, y) x4 + y? has no limit as (x, y) approaches (0,0).

A: Note: Since we can solve at most one problem at a time, we have solved the first one. Kindly…

Q: Express the limit as a definite integral. [Hint: Consider f(x) = x*.] lim n- 00 dx

A:

Q: 2. f(x) = Vx – 1 at x = 5

A: Function is continuous and differentiable both at x= 5.....for detail...go through the solution

Q: If f(x) is continuous at x=a , then f(x) is also differentiable at x=a.

A: " If F(x) is continuous at x=a then f(x) is differentiable at x=a." The condition for the function…

Q: Evaluate the following limit. sin xy + cos xy 2y lim (x,y) →(0,)

A: The objective is to evaluate the limit.

Q: Express the limit as an integral. * lim (5(x)2 - 2(x;)³)Ax over [0, 3] i = 1 dx

A:

Q: If functions ƒ(x) and g(x) are continuous for

A:

Q: Jaimie, a MAT student, claims that "For a continuous function f(x), defined on an interval [a ,6] ,…

A:

Q: Which of the following represents the slope of the tangent to f(x)=Vx+1 at x=8 ? Select one: 8+h-/8…

A: As we know that the expression for the slope of the tangent to the curve is:   limh→0fx+h-fxh…

Q: express the limit as a derivative.

A: Given expression

Q: What is the limit of (e^x-e^-x)/( sin (x)) as x approaches 0?

A: we have find the value of limx→0ex-e-xsin x

Q: Suppose f is continuous on [0, 1]. What is lim n→ ∞ 1 S fo 0 f(x) dx?

A:

Q: If ƒ is differentiable at a, must ƒ be continuous at a?

A: we know that for a given function if it is differentiable at given point then it must be continuous…

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A:

Q: Evaluate f x2 dx using the limit defintion of the definite integral.

A:

Q: Let f(x) be a continuous differentiable function that f(x)- 4 satisfies lim =D5 %3D x3 x-3 Which…

A:

Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f(x) = 2x 2 +…

A: Given:                fx=2x2+10x,     a=3 Find: f3+h and f3 f3+h=23+h2+103+h…

Q: Find the limit as x approaches c of the average rate of change of the following function from c to…

A:

Q: Which of following limits is the derivative of g(x) =5x at =2? 10 – 5h lim h 10 + 5h lim h>0 h 10 +…

A:

Q: Express the limit as a definite integral on the giveninterval.

A:

Q: . Explain why f (x) = sin (x² + 3r + 1) is a continuous function for all c eR.

A:

Q: Express the limit as a definite integral. [Hint: Consider f(x) = x".] lim 5 i = 1 1 dx

A: To express the limit as a definite integral, limn→∞∑i=1n3i4n5

Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

Q: If ƒ' is continuous on [0, ∞) and lim→∞ f(x) = 0, show that I p'(x) dx = = -f(0)

A: First we will integrate then put limit

Q: If f(2) is a continuous function at a point z, , then f(z) is differentiable at z, . But the…

A: Authoring guidelines:…

Q: that makes y continuous but not differentiable by writing the required procedures to that If: X≥0…

A:

Q: e* + e * — х2 — 2 lim —х sin?x – x2

A:

Q: dy -) using definition with limit of the following: dr 4. Find the differentiation (- y = /2x + 1.

A: We need to find the differentiation of y=2x+1 using the definition of limit. Now, according to the…

Q: 8) Based on the graph of y = g(t) below, state the value of each limit if it exists. If it does not…

A: Observing from the given graph we have

Q: Given a2 +1, x 1. Is f differentiable at x = 1? Is it continuous at x = 1?.

A:

Q: If ƒ is continuous on [a, b] and, what can you conclude about ƒ?

A: the integral given to us is ∫ab|f(x)|dx=0

Q: Assume that f(x) is everywhere continuous and it is given to you that f(x) + 8 lim x→2 x - 2 = -4 It…

A: Consider the given expression limx→2fx+8x-2=-4 Or, -4=limx→2fx+8x-2                            …1…

Q: Use a derivative (not L'Hospital's Rule) to evaluate (2 + h)4 – 16 lim h Enter the limit in the…

A:

Q: If f is a continuous function and F'(x)=f(x) for all x € R, then - x)dx=?

A:

Q: Express the limit as a definite integral on the given interval. lim S 19(x/)³ – 8x;]Ax, [2, 5] 1

A: The definition of definite integral is, ∫abf(x) dx = limn→∞ ∑i=1n f(xi*)△x   in the interval a, b…

Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

A:

Q: S (2t)*dt lim x→ =? х* — 1

A:

Q: For every continuous f:[0,+0) →R _function , if a>0 2fx'f(x*)dx =[ xf(x)dx Demonstrate that this…

A:

Q: Suppose that f (x) is a continuous function. 3 If | f(x) dx=5, then what is the value of z - f(x)]…

A: Given, ∫13f(x) dx =5Now,∫13[x -f(x)] dx= ∫13x dx -∫13f(x) dx =x2213 -5…

Q: d. Use the limit definition of the derivative, f (x) = lim f (x+h)-f(x) 1 +h)=j, to prove that VT =…

A:

Q: Suppose that f(x) is a continuous function on the interval [0, T] . If f(x)=(sin x)ª for 0<x<T, then…

A: we have to find the value of given function. the solution is following.

Q: If f' is continuous on [0, ∞) and limx→∞ f (x) = 0 , showthat

A: Given, If f' is continuous on the interval [0,∞) and limx→∞f(x)=0 show that, ∫0∞f'(x)dx=-f(0)

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: