Explanation Given: Need to find points with a horizontal or vertical tangent line on the curve with following parametric equation c ( t ) = ( t 3 − 4 t , t 4 − 12 t 2 + 48 ) Formula used: 1) Slope of the tangent line = d y d x 2) Chain rule states that d y d x = d y d t . d t d x 3) d d x ( k ) = 0 where k is any constant value 4) d d x ( x n ) = n x n − 1 Calculation: Let’s calculate equation of the slope for tangent line on given curve m = d y d x [Using formulae Slope of the tangent line= d y d x ] d y d x = d y d t . d t d x ... ( 1 ) [Using chain rule] c ( t ) = ( t 3 − 4 t , t 4 − 12 t 2 + 48 ) [Given] Assume that x ( t ) = t 3 − 4 t and y ( t ) = t 4 − 12 t 2 + 48 Since y ( t ) = t 4 − 12 t 2 + 48 d y d t = ( d d t ( t 4 ) − d d t ( 12 t 2 ) + d d t ( 48 ) ) d y d t = ( 4 t 3 − 24 t + 0 ) [Using formulae d d x ( k ) = 0 and d d x ( x n ) = n x n − 1 ] d y d t = 4 t 3 − 24 t Since x ( t ) = t 3 − 4 t d x d t = ( d d t ( t 3 ) − d d t ( 4 t ) ) d x d t = ( 3 t 2 − 4 ) [Using formulae d d x ( x ) = 1 and d d x ( x n ) = n x n − 1 ] On placing calculated value of d y d t and d x d t in equation ( 1 ) d y d x = 4 t 3 − 24 t 3 t 2 − 4 Hence calculated value of m with respect to t is as follows m = 4 t 3 − 24 t 3 t 2 − 4 Since tangent is horizontal, that is tangent is parallel to x axis, slope is zero for horizontal tangent. Replacing m by 0 in above equation 0 = 4 t 3 − 24 t 3 t 2 − 4 4 t 3 − 24 t 3 t 2 − 4 = 0 4 t 3 − 24 t = 0 [Since 3 t 2 − 4 ≠ 0 ] 4 t ( t 2 − 6 ) = 0 t = 0 , t = ± 6 t = 0 , t = 2...

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Chapter 12.1, Problem 78E

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The points with a horizontal or vertical tangent line on the given curve, and plot the curve for given parametric equation.

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Chapter 12.1, Problem 77E

Chapter 12.1, Problem 79E

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Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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The points with a horizontal or vertical tangent line on the given curve, and plot the curve for given parametric equation.

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The points with a horizontal or vertical tangent line on the given curve, and plot the curve for given parametric equation.

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