About Differentiation

Question

Here are a few of my questions! 

Thank you for taking the time to even click on this! 

 

1. Find the coordinates of the stationary points of the curve y= x(x-2)^3 and determine the nature of the points 

 

2, A curve has the equation y = 2x^3 - 5x + q, where q is a constant. 

   

   (i) If the tangent to the curve at the point where x = 1 has y-intercept equal to 7, find the value of q. 

   (ii) Find the equations of the normal to the curve which are parallel to the line x+y+5=0 

 

3. A liquid is poured on a flat surface. It forms a circular patch which grows at a steady rate of 5cm^2 / s. Find in terms of (MATH PIE),

 

(i) the radius of the patch 20 Seconds after pouring has commenced (I CAN SOLVE [i)...but i will put the question here just for context, thank you!!) 

(ii) The rate of change of the radius at this instant. 

 

 

Again, I would like to thank everyone who has taken a little bit of their precious time to read my question!! 

Answer 1

y = x(x-2)³

y' = 1(x-2)³ + 3x(x-2)² = (x-2)²[(x-2) + 3x] = 0

(x-2)²(2)(2x-1) = 0 has critical points when x=2 or x=½.

at critical number 2, the sign of y' is positive just to the left and also just to the right, so we have this behavior of y' over the numberline:

   on (-infinity,½)  y' is negative, y is decreasing

   on (½,2)          y' is positive, y is increasing

   on (2,infinity)   y' is positive, y is increasing

 

This information tells us that y has a local minimum at x=½.

 

I leave it to you to determine if there's an inflection point.  Also, you may find the y coordinates as needed.