Verify that the point P(a cos 0, b sin 0) lies on the ellipse y? = 1 62 a2 where a and b are the semi-major and semi-minor axes respectively of the ellipse . Find the gradient of the tangent to the curve at P and show that the equation of the normal at P is ax sin 0 – by cos 0 = (a² – b?) sin 0 cos 0. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a? > 262. If further, the tangent at P meets the y-axis at Q, show that a² |BQ| = 62 %3D

Q: Consider the parametric equations y = e' +t te [1, 10] Briefly explain how you know that the curve…

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Q: I of intersection. Show A graph both curves on 3- 2- -2

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Q: Find an equation, in slope-intercept form, of the line tangent to the curve with parametric…

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Q: x = 7 te', y = (t + 1)e', 0stsi Set up an integral that represents the area of the surface obtained…

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Q: Q. (a) Find an equation of the tangent plane to each surface at the given point. (b) Find symmetric…

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Q: A car travels over the hill having the shape of a parabola. When the car is at point A, it is…

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Q: A particle is moving along the curve given by the parametric equations x=1+t, y=1−t. Determine the…

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Q: Find (d^2)y/d(x^2) at t= pi/3 for the parametric curve given below. Write your final answer in the…

A: Given parametric curve, x = t+sin(t) y = t-cos(t)

Q: 2.) Given the parametric equations x=cos3t and y=sin3t. Find the equation of the normal line at the…

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Q: 4. Find the slope of the curve r = cos 20 at the origin 5. A particle moves along a path whose…

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Q: Find the slopes of the tangent lines to the ellipse 4x +y = 40 at y=2. The smaller slope is The…

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Q: Obtain the gradient to a curve with the parametric equations given by I = In - and y = sinh 2t whent…

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Q: Find the tangent line(s) to the parametric curve given by a = t¹ - 9t² and y=t² at (0, 9).

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Q: A mouse Q runs along the positive y-axis at the constant speed v at the origin. A cat P chases the…

A: It is given that∙Mouse Q runs along y-axis with a constant velocity v1∙Cat P runs along curve C with…

Q: 7. Calculate dy/dx at the point indicated: f(0) = (7tan O, cos O), 0=/4

A: Here we will use the concept of parametric equation and Derivative of parametric ewyations

Q: 5: A potato cannon launches a potato straight up from a height of 5 feet with an initial velocity of…

A: Step 1:Horizontal distance x(t)=ux​t+at2/2but the potato is launched vertically, so horizontal…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.