. Find the parametric equations for the tangent line to the Curve at the Specific point. X = e-t cos (t), y=e Sin(t), z = e"; (1,0,1) COS · Is the curve Smooth? If not, find the location which it isnt

Q: Find the exact value of 0.049768 x³y²z ds, where C is the curve with parametric equations x = e¯t…

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Q: Consider the parametric curve sin t + cos t, y = cos? t 0<t< 27 Find all values of t e [0, 27] at…

A: We find dx/dt and dy/dt

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: The given curve is rt=t2+1,4t12,et2-t. Notice that the given curve satisfies t=1(If t=1 is plugged…

Q: Consider the curve defined by the parametric equations: x=5t cos t at t = t y= 5t sin t Find the…

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Q: I the slope of the tangent line to the parametric curve X = 3 cos (t) y = 2 sin(t) ㅠ R 4 at the…

A: Given query is to find the slope of the tangent.

Q: Given the parametric equation x = t+ lnt and y = t - Int %3D dy d²y Find and dx da? Find the point C…

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Q: Find the length of the curve with parametric equations x = e' · cos t, y = e' sin t, with 0 <t <…

A: Given x = et.cos(t) y = et.sin(t)

Q: Find the parametric equations for the tangent line to the curve æ = t3 – 1,y = to +1, z = t° at the…

A: Find the tangent vector r't:                 r(t)=(t3-1)i +(t5+1)j +t5k Differentiate r(t):…

Q: Q2 - A) Find the parametric equations for the line that is tangent to the given curve : r(t) = (a…

A: To find The parametric equations for the line that is tangent to the given curve

Q: Given the curve with parametric equation x= e* and y= In(2-r²) Find the equation of the normal to…

A:

Q: 3. Consider the parametric curve C given by * = te', y =t + cos t. (a) Find the equation of the…

A: Given: The parametric equation of the curve C is  x=tet, y=t+cost (a) To find: The equation of…

Q: (a) Find the slope of the tangent line to the parametric curve x = 3cos t, y = 4sin t at t =…

A: Given, x=3cost,y=4sint at t=π4 and t=7π4. Differentiate the functions x and y with respect to t.…

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A:

Q: the length of the parametric curve a =t+cos(t), y =t – sin(t), 0 < t < 2n ?

A: The length of the curve formula is

Q: 3. Consider the parametric equations x = 1+ cos(t), y = -2+ sin(t), 0sts2M. 3.1. Find an equation…

A:

Q: Consider the parametric curve C that is defined by C: x=t^3, y=t^2+t, −1≤t≤3. (a). Find the…

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Q: Determine the exact length of the curve for the parametric equation x = tsint and y = tcost for 0 ≤…

A: To find the exact length (arc length) of the curve for the parametric equation x=t sin(t), y=t…

Q: Find parametric equations for the line that is tangent to the curve r(t) = e^t i + (sin t)j + ln (1…

A: The given curve is Differentiating with respect to t, we get Now, we find r(0) and r'(0) and

Q: 1. For the parametric equations: x = 4t3, y= 2 +60t – 8t?, find: a) All values of t at which the…

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Q: Find an equation of the tangent(s) to the curve with parametric equations tsint at the point…

A:

Q: Find the parametric equations of the tangent line LT to the curve C:r(t) = 2 cos(t)i + sin(t)j+ 2tk…

A: A parametric equation of a curve is given by: C: r(t) = 2cos(t)i + sin(t)j + 2tk Now, we have to…

Q: A curve is defined by the parametric equations cos z dz, sin z dz, t > 1. X= y= Find the length of…

A: Use the formula for the length of curve in parametric form.

Q: Let C1 be the curve with parametric equations x = cos t, y = sint – t, where t E [0, 5). (a) Find…

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Q: (d) Given r(t) = (ln(t+1), t cos 2t, 2¹). (i) Find r'(t). (ii) Find parametric equations for the…

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Q: 4. Determine the length of the parametric curve given by the following parametric equations. (a) x =…

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Q: 1. Eliminate the parameter to find a Cartesian equation of the curve given parametrically by (x(t),…

A: Recall: sin2t+cos2t=1

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: Here, the given parametric equation of the curve is: r(t)=(x(t),y(t),z(t)) where x=6et, y=t e2t, z=t…

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Q: Find parametric equations for the line that is tangent to the given curve at the given parameter…

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Q: The astroid curve below is defined by the parametric equations x(t) = 2 cos' t, y(t) =2sin't.…

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Q: Let C1 be the curve with parametric equations x = cos t, y = sin t – t, where t E [0, 5). (a) Find…

A: This question can be solved using the concept of equation of tangent and length of curve when the…

Q: (ь) Find the value of a for which the tangent to the parametric curve = 4t° +t? + at, y = t – 6 at t…

A: Given the two parametric curves x=4t3+t2+αt   and y=t4-6. We need to  find the value of α for which…

Q: Given the parametric curve x = t – sin(t),y = 1 - cos (t), find an equation for the line tangent to…

A: Given: x=t-sint y=1-cost at t=pi/3

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Q: Find parametric equations for the line that is tangent to the given curve at the given parameter…

A: Given equation rt=lnt7i+t-7t+8j+t×lnt7k......(1) Substitute the value of t=7,…

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A:

Q: Find the length of the parametric curve I = cos(t) y = 1 + sin(t) from t = 0 to t =

A: Given- x=cost and y=1+sint To Find- The length of the parametric curve from t=0 to t=π. Identity…

Q: Find parametric equations for the line that is tangent to the given curve at the given parameter…

A: Consider, r(t)=lnt4i+t-4t+8j+t lnt4kWe first evaluate r(t) at t=t0=4r(4)=ln44i+4-44+8j+4 ln44k…

Q: 1 '1+t²⁹ Given R'(t) = (4e²t, , 2cosh t), and R(0) = (1, -2,3).

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Q: x = +4r2 +4t +1 y =1+3 at the point P(1, 1).

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: Answer

Q: 2. Find parametric equations for the tangent line to the curve r(t) = (t8,t³,t7) at (1,1,1).

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Q: A curve is defined by the parametric equations T = sin t, y = 1 – cos t, 0<t< 27.

A: Given parametric equations is x = sin t; y = 1 - cos t, 0 ≤t≤2π b) Eliminate t and determine what…

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Q: Consider the following parametric equation. x=7(cos(t) + tsin(t)) y=7(sin(t) - tcos(t)) what is…

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Q: Graph the parametric curves x = cos t, y = sin t, 0 ≤ t ≤ 2π.

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Q: 1. Find the tangent to the curve given by the parametric equations: X = 1/t; y= VE e-t

A: Appliaction of derivative problem

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A:

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.