6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) = 0, f(1) = 1, and f(2) = 2. Show that there is a € (0, 2) such that f"(a) = 0.

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Q: 12.) Compute the first partial derivatives of the following function. f(x,y) = 2- tan 1 (7x² + 2y²)…

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Q: Find the best linear approximation to f(x) = x² on the interval [0, 2].

A: Here we use Taylor's approximation.

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A: Given query is to find partial derivative using definition.

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Q: 8. What is the relationship between a and b so that F(x, y) = (ary, 3ax + bry) is source free?

A: That's easy. Have a great day!!!

Q: Suppose F(x,y)=sin²(xy +e"), where x=u² +v and y = In(u² +v). Find the . ƏF ¸ ÔF value of when u =1…

A: The given function is: F(x,y)=sin2xy+eywhere:x=u2+v         and         y=lnu2+v

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Q: 8) Find the linearization of f(x) = 6 + X+1 tan 4 dt at x = 0. 1.

A: On solving this we will get

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Q: Compute the partial derivative fxxyz for f(x, y, z) = cos(cos(x3 + z2 )) + x2 ln(y2 + z2 ).

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Q: Compute the given partial derivative. f(x, y) = 8 sin(4x² – y), (0, 1) f,(0, 1) = %3D

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A: Given,A(x)=∫0x f(t) dt

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Q: Check whether the given function is satisfying the Fubini’s theorem about mixed partial derivative…

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Q: 1, t 1 differentiable at t = 1.

A: As per bartleby guidelines we only give first question answers.

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Q: 2. Show that the function psin(2r) e-*dt g(x) = is differentiable on R and compute its derivative.

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Q: 1. Integrate the function f(x, y) = x – y over the triangle T with vertices (0,0), (1,0), and (1,2).

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A: Prove that ∫abx-ax-bf"xdx=2∫abfxdx.

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Q: Suppose that f: R2R is a function given by S x²+y² f(x, y) 0 for xy = 0. Show that f(x, y) is…

A: We have to show the given function  f(x, y) is continuous  at ( 0,0)

Q: 2. i) Give an example of a nonzero continuous function f on [0,1] such that f(x)dx = 0.

A: We have to give an example of a nonzero continuous function f  on [0,1] such that  ∫01f(x)dx = 0

Q: Let f be a function with continuous partial derivatives and defined by: z=x2*f3(xy,y2) If taken u…

A: We will find out the required value.

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Q: 4. Use the limit definition of partial derivative to find f, (x, y) and f, (x, y). a) f(x,y)=/x+y

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Q: Letv = x2sin(v) + vexy, where X = s + 2t and y = st-

A: Solution is below.

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.