how do I know whether I need to integrate or differentiate an equation?
Remember that when you differentiate an equation, you are finding the rate of change. If you need to find a rate of change, you differentiate the equation.
E.g. In your first example, you need to find the rate of change of no. of spectators, so you would differentiate the equation for the no. of spectators.
If you have a rate of change and you need to find a variable, you integrate the equation.
E.g. If you were given the equation for acceleration of an object and you needed to find the velocity or displacement, you would integrate the equation, because acceleration is a rate of change of velocity, and velocity is a rate of change of displacement.
Lets look at your examples.
You need to find the constant k, and you know that growth is fastest when t = 4. The word “growth” gives you a clue that you need to differentiate, as differentiation is the rate of change, or growth. The rate of change of the number of spectators can be modelled by:
This equation describes how the amount of spectators changes over time. Then, we know that this growth is fastest when t = 4. Finding the maximum and minimum of a function is another reason why we apply differentiation - points of maximum and minimum have a gradient of zero (the derivative of a function at these points is equal to zero):
Therefore, we need to find the derivative of the function representing the growth of the amount of spectators - the rate of change of the growth:
We know that when t = 4 then P’’ = 0:
In your second example, you need to find the maximum height of the ball above the ground. The word “maximum” is a clue that you need to differentiate the function. Then you can find the time when the derivative is equal to zero:
That means that in 2.3sec the ball will reach its maximum height. Then we can substitute this time into your original equation to find this height: