X2 = 3 + (-t)' y2 = (-t)' + 1 ? Find equation for the...

x_2 = 3 + (-t)' \newline y_2 = (-t)' + 1 \newline \newline ? Find equation for the tangent line at t=0 \newline Write the equation of tangent line in two forms. \newline (1) Slope intercept \newline (2) Using parametric equation

Transcript

00:01 We will now determine the parametric equations of the line which is tangent to the given curve equation at their points t not equal to zero for the first curve in part a and for the second curve in part b t not equal to two so let's do the part a i have copied the curve equation as like this that is in component form like this this basically represents the x of t or we can simply write down this as x and this represents x as x and this represents represents the y of t and then the next two component represents z of t so first let's determine the point by plugging t0 equal to 0 or t equal to 0 into the each of the points so that we get the point first so i'm going to plug in t equal to 0 into each of this x t y times the which means this will be sine 0 and this will be 0 squared minus of a cost 0 and this will be e power 0 so we get a sine 0 is 0 and this is 0 squared minus cos 0 is 1 so it is negative 1 and equal 0 is 1 so basically we get the point this is the point on the curve equation so we have determined the point first now let's find the derivative of the curve equation so therefore i'm going to find r prime of t so we have to find the derivative of each of the parts first let me do for sine t differentiation of sine t is cost t then i do the derivative of t squared minus cost t so this is equal to 2 t when apply the power rule minus derivative of cost is negative sine t we already have a negative so this becomes a positive sine t then derivative of epo t is e power t so this basically represents the r prime of t now we have to plug in this value of t equal to 0 into this so that we get r prime of 0 so let me do this this or prime of 0 equal to cost 0 comma 2 times of 0 plus sine 0 comma e power 0 so let's evaluate this this is equal to cause 0 is 1 and 2 times 0 is 0 and sine 0 is also 0 so this comes out to be 0 and epo 0 is 1.

02:31 So this basically represents r prime of t at for the t value 0.

02:38 Now to determine the parametric equations, we write down like this, that is x of t, y of t, and z of t.

02:49 Then we put the corresponding point in each of this parameter, that is the, this is the point 0 minus 1 and 1.

02:58 So we put like this 0 minus 1 and 1 and then we put plus we have to put the component or the coefficient for the t.

03:09 So and for that we have to use the r prime of 0...

Question

x_2 = 3 + (-t)' \newline y_2 = (-t)' + 1 \newline \newline ? Find equation for the tangent line at t=0 \newline Write the equation of tangent line in two forms. \newline (1) Slope intercept \newline (2) Using parametric equation

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