Answer:
(a) t = ±2
(b) t ∈ {0, 1}
(c) In navigation terms: east by north. The slope is about 0.42 at that point.
Step-by-step explanation:
(a) dy/dx = 0 when dy/dt = 0
dy/dt = 3t^2 -12 = 0 = 3(t -2)(t +2)
The slope is zero at t = ±2.
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(b) dy/dx = (dy/dt)/(dx/dt) = <em>undefined</em> when dx/dt = 0
dx/dt = 6t^2 -6t = 6(t)(t -1) = 0
The slope is undefined for t ∈ {0, 1}.
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(c) At t=3, dy/dx = (dy/dt)/(dx/dt) = 3(3-2)(3+2)/(6(3)(3-1)) = 15/36 = 5/12
The general direction of movement is away from the origin along a line with a slope of 5/12, about 22.6° CCW from the +x direction.
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The first attachment shows the derivative and its zeros and asymptotes. It also shows some of the detail of the parametric curve near the origin.
The second attachment shows the parametric curve over the domain for which it is defined, along with the point where t=3.
Well, first let's identify which answers are incorrect, then it will be easier to figure out which are correct.
A. Equilateral: An equilateral triangle is a triangle with 3 equal sides. Since there are 180 degrees in a triangle, an equilateral triangle would have three sides of 60 degrees, and none of 45 degrees. Answer? Incorrect.
B. Isosceles: An isosceles triangle has two sides that are equal. 45 and 45 are equal, therefore, this answer is: Correct!
C. Scalene: A scalene triangle has three unequal sides, therefore, this answer is incorrect.
D. Obtuse: An obtuse triangle has one angle that is more than 90 degrees, therefore, since 45 and 45 equal 90 already, this answer is: incorrect.
E. Right: A right triangle has one right angle (angle that equals 90 degrees) since 45 + 45 = 90, and 90 + 90 = 180, this answer is: Correct!
F. Equiangular: This last choice is practically the same as the first, therefore the answer is: incorrect.
The two correct answers are: B Isosceles, and E Right!
First you distribute the -12 to the (x+5) getting 144=-12x-60.
Next you add 60 to each side getting 204=-12x
Then you divide each side by -12 getting -17=x, which is the answer.<span />
