limits and zero problems

Question

Answer 1

(7)

The position of the particle is x(t) = t² -6t +5

The velocity of a particle can be found out by taking first derivative of x(t) = t² -6t +5 .

Velocity v(t) = x' (t)

The time at which the velocity of particle zero is 2t - 6 = 0

 2t - 6 = 0

2t = 6

t = 3

The time at which the velocity of particle zero is t = 3.

Answer 2

(8)

Factoring out the terms x³ in numerator and x² in denominator .

Substitute x = - ∞ in the limit 

Answer 3

(9)

The function is xy² - 2y + 4y³ = 6

To find out the slope of the function , first we need to find out it's first derivative .

Find out the value of x when y = 1 .

x(1)² - 2(1) + 4(1)³ = 6

x - 2 + 4 = 6

So x = 4 when y = 1 .

Derivative the function with respect x using implicit derivative technique .

xy² - 2y + 4y³ = 6

The slope of the function at point (4 , 1) .

The slope of  xy² - 2y + 4y³ = 6  at  (4 , 1)  is  -1/18