Find formulas for the distances between (0, 0) and (a, a2 ), a > 0, along the line between these points and along the parabola y = x2 ? Make a conjecture about the difference between the two distances as a increases?

Q: ind the point P on the line y = 5x that is closest to the point (78,0). What is the least distance…

A: This question is related to straight line. we will solve it using the condition of shortest distance…

Q: Find the tangent/s as directed to the parabola (x + 2y)2 + 2x – y – 3 = 0, parallel to the line x –…

A: The objective is to find the equation of tangent line to the parabola x+2y2+2x-y-3=0 which is…

Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 ≤ t ≤ 2π, closest to the point (3/4, 0).…

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Q: The cubic y = ax3 + bx? + cx + d tangent to the line y = 12x + 13 at (-1, 1) and have a horizontal…

A: Solution...

Q: Find the tangent/s as directed:to the parabola (x + 2y)^2 + 2x – y – 3 = 0, parallel to the line x –…

A: Given equation of line is  x-4y=24y=x-2 y=14x-12 So comparing with the slope equation formula,…

Q: Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approxim

A: Given,         f(x, y)=11-2x-y

Q: The tangent of the parabola x = y² + y + 1 at the point (1,0) is

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Q: Find the x Coordinate of the point on the graph of 4=x² where the to the Secant. Longent tine.…

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Q: Find the length of the following curve. y = 2x + 4 on 3-2, 24 (Use calculus.)

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Q: Find the arclength of y = 2x2/3 from x is greater than or equal to 1 and less than or equal to 2

A: The arc length of the function y=2x23 between x=1 to x=2.

Q: 8. At what point of the parabola is the tangent line parallel to 7x+y-2=0? y = 2r-3x +1 O (1,0) O…

A: Here is the solution:

Q: Find the point Q that is on the curve nearest %3D the point (4, 1)

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Q: Find the equahion a Hhe tangent Lie FlX) =482- 5X +7 when R=1

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Q: For what values of a and b is the line 2x + y = b tangent to the parabola y ax? when x = 2?

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Q: Find the tangent plane to the equation z = - 32² – y² + 3y at the point ( – 5, 3, – 75)

A: z=-3x2-y2+3y  and the point (-5,3,-75)

Q: Find the point on the Curve y rx that is closest to the point (2,0)

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Q: The slope of the tangent line to the parabola y = 4x? + 7x + 2 at the point (– 4, 38) is: The…

A: Equation of parabola,  y = 4x2 + 7x + 2 Given point is (-4, 38)

Q: Find the length of the arc of the curve 4y2 = 9x3 from the origin to the point (2,3v2).

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Q: Show that the square of the length of the subtangent at any point on the curve by2=(x+a)3 (b is not…

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Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 <=t<= 2pai closest to the point (3/4,…

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Q: The area bounded by the parabolas (x − 2)^2 − 9py=0 and y^2=9p(x−2) is to be revolved around x = 2.…

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Q: Find the point (xo, yo) on the graph of y = x + 3x² + 4x – 5, at which the slope of the tangent line…

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Q: and the line perpendicular to Find the area of a triangle bounded by the y axis, the tine f(z) - 9 -…

A: We have to find the area of the region bounded by the y-axis, the line f(x)=9-67x, and the line…

Q: Find the tangent/sto the parabola (x + 2y)2 + 2x – y – 3 = 0, parallel to the line x – 4y = 2.

A: The equation of the parabola is given as  x+2y2+2x-y-3=0. On simplifying,…

Q: The x-coordinate of the point where the tangent line to the parabola y=2x^2 at x=1 intersects the…

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Q: 7. For whit value of b will the line y =- y 3x +4x -1? 2x+ b be tangent to the parabola

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Q: Find the tangent plane to the equation z 5a2 - 4y +5y at the point (5, 5, - 200)

A: The equation of the tangent plane to a surface  is given by fx(x-a)+fy(y-b)+fz(z-c)=0where (a,b,c) =…

Q: Show that the square of the length of subtangent at any point on the curve by 2 = (x+a)3,(b is not…

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Q: Find the point (xo, yo) on the graph of y = x³ + 3x? + 8x – 3, at which the slope of the tangent…

A: To find the slope of the function we need y'y=x3+3x2+8x-3y'=3x2+6x+8

Q: 8.) Find the exact length of the curve y= over the interval (0.4]. 2 %3D -

A: Given,          The exact length of the curve y=x22 over the interval 0,4 .

Q: At what point on the parabola y =  3 x2+ 2x is the tangent line parallel to the line y = 10x −2? Use…

A: First prinicipal : dydx=limh→0f(x+h)-f(x)h

Q: Determine the coefficients a, b, c and d so that the curve y = ax* + bx³ + cx² + dx + e will pass…

A: Since we have to find value of a,b,c and d

Q: The line that is normal to the curve x + 2xy - 3y = 0 at (4,4) intersects the curve at what other…

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Q: on of the tangent to y = 3e" + 3ex that is parallel to

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Q: The slope of the tangent line to the parabola y 4x? + 7x + 3 at the point (0, 3) is: The equation of…

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Q: The slope of the tangent line to the parabola y 3x? + 2x + 3 at the point (– 5, 68) is: The equation…

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Q: Find x for which the line tangent to the parabola y = 4x − x^2 is horizontal

A: The objective is to find x for which the line tangent to the parabola y = 4x − x^2 is horizontal.

Q: Given the parabola: x2+12y=0 and the tangent line x+y-3=0, find the point of tangency

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Q: Find the length of the curve. y = In(secx), 0Sx</4

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Q: 1. Tangent to the parabola z = x + 3y?, x = 1, at point (1,2, 13

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Q: The semi-hyperbola H(X)=(sqrt(x^2-1)) is continous at which x interval?

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Q: What is the value of c such that the line y=2x+4 is tangent to the parabola y=cx^2.

A: Given that the line y=2x+4 is tangent to the parabola y=cx2. The derivative of y=cx2 is dydx=2cx.…

Q: find the slope of the curve at the point indicated. 19. у %3D 5x — Зх?, X =

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Q: Find the arclength of y = 4x2 - 4 where x is greater than or equal to 0 and less than or equal to 4

A: Concept Used: The arc length of the function is the length of the curve y=fx on the interval a,b is…

Q: In the xy-plane, the line x + y = k, where k is a constant, is tangent to the graph of y = x^2 + 3x…

A: Given line is x + y = k, where k is a constant it is tangent to the graph of y = x^2 + 3x + 1…

Q: integrate ƒ over the given curve. ƒ(x, y) = x2 - y, C: x2 + y2 = 4 in the first quadrant from (0, 2)…

A: The given curve C is a circle with center at origin and radius of 2 units. Let the parametric…

Q: Find the length of the curve 3y = (x² + 1) ? from x = 0 to x = 3.

A: Given:

Q: 10. At what point on the parabola y = 3x² + 2x is the tangent line parallel to the line 10x – 2? y

A: Given,          The parabola y=3x2+2x is the tangent line parallel to the line y=10x-2.

Q: At what point on the parabola y=3x^2+2x is the tangent line parallel to the line y=10x-2? Use first…

A: We have to find the point.

Q: where c is a constant, then prove that the foot of perpendicular from the origin upon the straight…

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.