Question

To make a phrase/words, solve the 18 application of derivatives below.  Then replace each numbered blank with the letter corresponding to the answer for that problem. Show all solutions on the answers given below.

"   __ __ __ __ __ __ __       __ __ __ __      __ __ __ __ __ __ __      __ __ __ __ __      __ __ __ __ __"       

"__ __ __ __ __ __ __      __ __      __      __ __ __ __ __ __ __      __ __ __ __ __      __ __ __ ."     

Derivative Application Problems:      

1.  Find the equation of the line normal to the curve  f(x) = x³ – 3x²  at the point (1, -2).          

2.  Find the equation of the line tangent to the curve  x² y – x = y³ – 8  at the point where x = 0.          

3.  Determine the point(s) of inflection of  f(x) = x³ – 5x² + 3x + 6.         

4.  Determine the relative minimum point(s) of  f(x) = x⁴ – 4x³.        

5.  A particle moves along a line according to the law s = 2t³ – 9t² + 12t – 4, where t≥0 .  Determine the total distance traveled between t = 0 and t = 4.             

6.  A particle moves along a line according to the law s = t⁴ – 4t³, where t≥0.    Determine the total distance traveled between t = 0 and t = 4.        

7.  If one leg, AB, of a right triangle increases at the rate of  2 inches per second, while the other   leg, AC,  decreases at 3 inches per second, determine how fast the hypotenuse is   changing (in feet per second) when AB = 6 feet and AC = 8 feet.     

8.  The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is leaking   out at the rate of ½ cubic inch per second.  Determine the rate (in inches per second) at which  the water level is dropping when the diameter of the surface is 2 inches.     

9.  For what value of y is the tangent to the curve  y² – xy + 9 = 0 vertical?        

10.  For what value of  k  is the line  y = 3x + k tangent to the curve  y = x³ ?        

11.  Determine the slopes of the two tangents that can be drawn from the point (3, 5) to the parabola    y = x² .              

12.  Determine the area of the largest rectangle that can be drawn with one side along the x-axis and  

two vertices on the curve  y = e^-x2

13.  A tangent drawn to the parabola  y = 4 – x²  at the point (1, 3)  forms a right triangle with the   coordinate axes.  What is the area of this triangle?         

14.  If the cylinder of largest possible volume is inscribed in a given sphere, determine the ratio of the   volume of the sphere to that of the cylinder.

15.  Determine the first quadrant point on the curve  y²x = 18 which is closest to the point  (2, 0).     

16.  Two cars are traveling along perpendicular roads, car A at 40 mph, car B at 60 mph.  At noon when   car A reaches the intersection, car B is 90 miles away, and moving toward it.  At 1PM, what is   the rate, in miles per hour, at which the distance between the cars is changing?

Answer 1