Module 4 Review

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Module 4 Review Parametric and Vector Calculus J

1. A particle moves in the xy-plane such that its position for time đ?‘Ą ≼ 0 is given by < 3đ?‘Ą & − 19, đ?‘’ &,-. >. What is the slope of the tangent to the path of the particle when đ?‘Ą = 4? 23 Slope of the tangent means ! 24 đ?‘‘đ?‘Ś đ?‘‘đ?‘Ś đ?‘‘đ?‘Ą 2đ?‘’ &,-. = = đ?‘‘đ?‘Ľ đ?‘‘đ?‘Ľ 6đ?‘Ą đ?‘‘đ?‘Ą Then, evaluate the derivative at đ?‘Ą = 5: 2đ?‘’ ; đ?‘’ ; = 30 15

2. The position of a particle in the xy-plane is given by the vector < ln đ?‘Ą , 5đ?‘Ą & + 11 >, for đ?‘Ą > 0. Write, but do not evaluate, an integral expression that represents the total distance the particle travels from đ?‘Ą = 2 to đ?‘Ą = 6. .

;

1 đ?‘Ą

&

+ (14đ?‘Ą)& đ?‘‘đ?‘Ą

3. A plane curve is represented by the parametric equations đ?‘Ľ đ?‘Ą = đ?‘Ą & and đ?‘Ś đ?‘Ą = đ?‘Ą A + 3đ?‘Ą & . Write an expression for the rate of change of the slope of the tangent to the path of the curve at time t. 2B3

Rate of change of the slope means B ! 24 đ?‘‘đ?‘Ś 4đ?‘Ą ; + 6đ?‘Ą = = 2đ?‘Ą & + 3 đ?‘‘đ?‘Ľ 2đ?‘Ą đ?‘‘ & & đ?‘‘ đ?‘Ś đ?‘‘đ?‘Ą (2đ?‘Ą + 3) 4đ?‘Ą = = = 2 & đ?‘‘đ?‘Ľ 2đ?‘Ą 2đ?‘Ą

5. A particle moving in the xy-plane has position given by the vector < � &, , � > for � ≼ 0. (a) Write, but do not evaluate, an expression that represents the speed of the particle at time t. �� ��

&

đ?‘‘đ?‘Ś + đ?‘‘đ?‘Ą

&

=

(2đ?‘’ &, )& +

1 2 đ?‘Ą

&

(b) State the acceleration vector of the particle at time t = 1. 1 &, đ?‘Ž đ?‘Ą =< 4đ?‘’ , − > ; 4 đ?‘Ą 1 & đ?‘Ž 1 =< 4đ?‘’ , − > 4

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