Find an equation for the line tangent to the curve at the...

Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d^2y / dx^2 at this point. x = sec^2 t - 1, y = cos t; t = ?/3 Write the equation of the tangent line. y = [ ]x + [ ] (Type exact answers, using radicals as needed.)

Transcript

00:01 In this question, we have to find the equation of tangent line given x is equal to sec squared t minus 1 and y is equal to cost t we have to find the value of tangent line equation at t is equal to pi upon 3 therefore x at t is equal to pi upon 3 check square pi upon 3 minus 1 is equal to check square pi by 3 is 2 2 square minus 1 is equal to 3 and y is equal to cost pi by 3 is equal to cost pi by 3 is equal to 1 upon 2 now d y by d f s is equal to d .y by dt upon d x by dt is equal to d by dt of cost t upon d by dt of sake squared t minus 1 is equal to minus sine t upon 2 into sick t into 10 t is equal to minus sine t upon 2 into sick t into 60 into 10 t is equal to minus minus sine t upon sec squared t 2 sec squared t into 10 t is equal to sine t upon cost t which is equal to minus 2 into cancel common term sine t upon sine t from numerator and denominator 1 upon sec square t is equal to cos square t into cost t is equal to minus 1 upon 2 therefore, d.

03:22 Y by dx is equal to minus 1 upon 2 cost cube t.

03:38 D .y by t x at t is equal to pi by 3 is equal to minus 1 upon 2 cost cube pi by 3 is equal to minus 1 upon 2.

04:11 Minus 1 upon 16 therefore equation of tangent line is y minus our point is x is equal to 3 and y is equal to 1 upon 2 therefore equation of tangent line is y minus 1 upon 2 into tangent at dy by d x at t is equal to pi by 3 minus 1 upon 16 into x minus 3 y is equal to minus x upon 16 plus 3 upon 16 plus 1 upon 2 y is equal to minus x upon 16 plus 3 plus 8 therefore y is equal to minus x upon 16 plus 11 upon 16 this is our required equation of tangent line now, d .y by dx is equal to minus 1 upon 2 cost cube t.

06:17 D .y by d x, dway by d x is equal to minus 1 upon 2.

06:44 Substitute, cost t is equal to y...

Question

Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d^2y / dx^2 at this point. x = sec^2 t - 1, y = cos t; t = ?/3 Write the equation of the tangent line. y = [ ]x + [ ] (Type exact answers, using radicals as needed.)

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