Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applied max/min questions.
Type 5: Rates - Test Yourself 1.


 

Time.

1.

 

Step 1: Find the distance AC. Just use Pythagoras' Theorem:

Step 2: Determine the time to swim the distance AC using the relationship Distance = Speed × Time.

Step 3: Determine the time to run from C to D which is a distance
of (3 - x) km:

Step 4:

Step 5: Differentiate etc:
  2.
  3. (i)

 

(ii)

Test to show change of gradient:

a 1 1.155 1.5
T' -0.01 0 0.2

Change in gradient from negative to positive - hence a minimum time taken when a = 1.155 km.
   
   
   
Costs. 6. (i)

(ii) Minimum cost = $304.53.
(Clearly the cost takes in maintenance, actual costs, depreciation, etc).

(iii) The time for the journey is 3.39 hours. The Road Safety campaigns urge drivers to STOP, REVIVE, SURVIVE after about two hours. Hence the 278 km journey should be made in two stages.
  7.
 

8. (i)

(ii)