1. In this problem you will prove that the shortest distance between two points is a line using the calculus of variations. Consider a curve y(x) whose endpoints are fixed: y(x = 0) = 0, y(x = d) = h. (1) That is, the curve goes from (0,0) to (d, h). (a) Write an integral expression for the arc length l of the curve in terms of the function y(x). Your expression should have the form = | dxo(y(x),y'(x)). (2) Determine the function o. Hint: Use the fact that the infinitesmal distance ds between two Vdx? + dy?. Write this in terms of dr points (x, y) and (x + dx, y + dy) is given by ds and the function y(x).

Q: 4. Fond theequation ofthe tangent line to the graph of the Function FCx)=R-JX at the point (1,0).

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Q: (b) Find a , b and c if the curvey = x + a x + b and y =c x + x have tangent lines paralle to each…

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Q: 5. Find the equation of the tangent line to the curve of the function Y= 2x-x +2 at the point (1,3).

A: Given,  y = 2x3-x2+2   and the point 1, 3 We have to find the equation of the tangent line at the…

Q: Prove that the curves y=f(x),(f(x)>0) and y = f (x)sin x, where f(x) is differentiability function,…

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Q: Find the point on the curve y = x-11x+5 at which the tangent has the equati on y = x - 11.

A: We will find the  point on the curve at which the equation of tangent is given as following.

Q: 5) Determine the slope of the tangent line to the function with the given information: a. f(x, y) =…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three subparts…

Q: 10. y f(x) -2 -2+ a. f(-4) b. f(0) c. f(2) d. f(4) 4, 2.

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Q: The tangent line to y = f(x) at (9,-10) passes through the point (6,-5). Compute the following. a.)…

A: Given query is to find the value of function and derivative at x = 9.

Q: If √x + √y = 9 and y(4) = 49, answer the following: a) y b) The slope of the tangent line to the…

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Q: Therefore, the curve y = x* - 2x2 + 2 has horizontal tangents at x-0, 1, and -1

A: Just compute the derivative and equate it to zero.

Q: 10) Find the tangent line to the curve 16 1 at the point (4 23).

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Q: (a) Find the slope of the curve at point P and write the equation of tangent line at point P. 2x2…

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Q: John Wallis also invented a method of finding the tangent line, which we describe as follows: Given…

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Q: Tangents are drawn to x² + y² = 1 from any arbitrary point P on the line 2x + y - 4 = 0. The…

A: Tangents are drawn to x2+y2=1 from any arbitrary point P on the line 2x+y−4=0. We have to find the…

Q: Use Weiestrass's definition of limit to show that lim (7x - 3)= 32. x-5

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Q: The equation of the tangent to the curve y = x – 3x² – 2x + 2 at the point where the curve cuts the…

A: We have to find  the equation of tangent in the form of y=mx+c.