Decide if the statement is true or false, and select the correct explanation. Ifa < b andf'(x) is positive on [a, b] then f(a) < f(b).

Q: Let c>0 be a constant.  Is this result true or false?

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Q: Find constants c1 and c2 such that F(x) = c1x sin x + c2 cos x is an antiderivative of f (x) = x cos…

A: To find constant c1 and c2 such that Fx=c1 xsinx+c2cosx is an antiderivative of fx=x cosx. Solution:…

Q: Find f(x) if f°(x)=cos x with f(0) =1, f'(0) = 2 , and f'(-)=3.

A: Keep on integrating we are end up with an answer. For those constants we need these initial…

Q: If f(x.y) = In(x² + y²) + find fr and fy -

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Q: Find constants c1 and c2 such that F(x) = c1xex + c2ex is an antiderivative of f (x) = xex .

A: Given: Fx=c1xex+c2ex is an antiderivative of fx=xex To Find: c1=? and c2=?

Q: Find the linearization of f(x) = (1+ x)20 at x = 0. Use it to approximate (1.001)20.

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Q: 3 Let f(x) = 2 + In(bx), where b is a positive constant. Find f'(x).

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Q: Prove that = va² – x* if y =(f)sin~+ () + (9)v&* - x²

A: Differentiation wrt x

Q: Given f (x,y) = x³ + y3 – 3y² – 3x² – 9y, find the critical points and use the Second Derivative…

A: solution :-

Q: Find f'(z) of f(z) = e(z^3) - 5z.

A: Given :- f(z) = e(z^3) - 5z

Q: of and ду of For the function f(x,y) = In (x+ 3y), find

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Q: If f(x,y)=x²y+y sin x, Prove that дхду дудх

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Q: Assume that f(x) is a differentiable and invertible function so that f(3) = 4 and f'(3) = .…

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Q: Let f(x) = 28x6. Find the most general antiderivative. F(x) :

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Q: If f"(x) = V - cos(x). f'(0) = 2, and f(0) = 0, find f(x). %3D

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Q: Find the absolution extrema of f(x) = (In x)2 on [1, e].

A: We have to determine absolute extrema of girls question

Q: If f(x1, x2) = x1 sin(x2)e¬#, what is the gradient V f(x) of f? Recall that V f(x) ´Əx,f

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Q: \+etari), find f.. S, and f.- xyz 3. If f(x,y,z)= hn 2

A: We have to find fx , fy , fz.

Q: Show that f (x) = 1 − |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive…

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Q: Suppose that C is a constant. Let Cr +1 f(x) = 7 +I+1° - , x E R.

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Q: Where a is a constant, then a is equal to:

A: Since, the joint probability distribution function is given .. Here, wa want to the value of 'a'

Q: Find the antiderivative F of the function f(x)= x^2 + 2x + 2 such that F(-1) = -1.  F(x) =

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Q: Find the antiderivative of f(x). f(x) = 3e6x+ sec2(x) +x3

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Q: If G(x,y) is differentiable with respect to x and y then it is always possible to find the number…

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Q: Find f, and f,. f(x, y) = 13x In xy

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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: Differentiate the function. y=In(x²+e' y=In(x²+ean x)

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Q: For f(x, y) = 2y3 + x² – 4x – 4xy – 9, find the critical points and use the - Second derivative test…

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Q: Find the linear approximation of f(x) = (5 + x)n at x = 0. Use the approximation to estimate the…

A: fx=5+xn at x=0. Differentiating with respect to x, we get f'x=n5+xn-1 Now, f0=5+0n=5n &amp;…

Q: Find of the function f(x,y) = sin(x²y³). əx3

A: The product rule of differentiation : (PQ)'=PQ'+QP' sinx'=cosxcosx'=-sinxxn'=nxn-1 Use the given…

Q: Find f(x), the most general antiderivative of the function: f' (x) = 3x –

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Q: Find the linear approximation to f(x) = (x+2) at 0 and use it to approximate (2.001)

A: Given :           f(x) = (x + 2)5 at 0 To find :             The linear approximation of f(x)…

Q: Suppose f'(x) = c for all x, where c is a constant. Show that, for some constant d, f(x) = cx + d.

A: suppose f'(x) =c for all x, where c is constant .Show that for some constant d, f(x)=cx+d

Q: If u = f(x), then f- dx ++2x+1 +C for some constant k. Compute kf(1).

A: In this question we have to compute kf(1).

Q: a.) If     f(x) dx   = 4x9 − 8x6 + C, find f(x). f(x) =

A: Given: ∫fxdx=4x9-8x6+c                                          ..1 for finding fx, we differentiate…

Q: Given the function y=f(x), y" is found to be positive over the interval (4,8), then over this…

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Q: Find the linearization L(X) of the function at a. f(x) = Vx, a = 16 L(x) =

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Q: If g(g(x)) is continous at 0, then g(x) is continous at 0.  True or False?

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Q: Find the linear approximation L(x)of f(x)=x^4 at the point a = −5 L(x)=

A: Given that f(x) = x4 We have to find the linear approximation L(x) of f(x)=x4 at the point a= -5.

Q: Find the linearization of L(x) of f(x) at x=a f(x)=StartRoot x2 +17, End Root a =8

A: Given function and point is

Q: Find the linearization L(x) of the function at a. п f(x) = sin(x), a = 6 L(x) =

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Q: Find all the x-intercepts of f(x)=cos(x) on the interval [0,2π)

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Q: Find the most general antiderivative of f(x) = -10x – (1 – x²)-1/2, where -1< x < 1.

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Q: For f(x, y), find all values of x and y such that f,(x, y)= 0 and f,(x.y)= simultaneously.

A: Please refer to the image below

Q: 2R then the constants R TCX If f(x) Rsin .2 S,f' and f(x)dx and S are respectively.

A: Introduction :Here we have asked to find the value of R and S

Q: Find constants c1 and c2 such that F(x) = c1xe -x + c2e -x is an antiderivative of f(x) = 3xe -x .

A: Given: Fx=C1xe-x+C2e-x  and fx=3xe-x

Q: Find L(x) (Linearization function), given f(x) = ev2x at the value a = 2.

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Q: Prove that (x – 2)dx = – f,(x – 2)dx |

A: To prove that:∫13x-2dx=-∫31x-2dx

Q: If f is a quadratic function such that f(0) = 9 and f(x) x2(x + 1)3 dx is a ratianal.functian,,find…

A: This question is about finding f'(x)  at x=0  where f(x) is a rational function

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.