Final answer:
The rate of change of G(t) = 90 + 45cos(t²/18) from t=0 to t=8 is found by taking the derivative G'(t), using the chain rule for trigonometric functions, and then evaluating it at various points in the interval.
Step-by-step explanation:
The student is asked to calculate the rate of change of the function G(t) = 90 + 45cos(t²/18) on the closed interval 0 ≤ t ≤ 8. This involves finding the derivative of the function, known as G'(t), and then evaluating it at various points within the given interval.
Steps to Calculate the Rate of Change
Take the derivative of the function G(t) with respect to t. This will give us the rate of change of G at any point t.
Since G(t) involves a trigonometric function, we will use the chain rule. The derivative of cos(u) with respect to u is -sin(u), and by applying the chain rule, we have d/dt[cos(t²/18)] = -sin(t²/18) * d/dt[t²/18].
Finally, we find the derivative of t²/18 with respect to t, which is (2t/18). So, the complete derivative G'(t) is -45sin(t²/18) * (2t/18).
Evaluate G'(t) at points along the closed interval from 0 to 8 to determine the rate of change at these points.
To calculate the rate of change of G on the closed interval 0<=t<=8, we need to find the derivative of G(t) with respect to t. The derivative of 90+45cos(t²/18) is -1/18 * sin(t²/18) * 2t, which simplifies to -2t/18 * sin(t²/18).
Now we can substitute the values of t from 0 to 8 into the derivative to find the rate of change of G over that interval. For example, when t = 0, the rate of change of G is 0, and when t = 8, the rate of change of G is -16/9 * sin(64/9).
So, the rate of change of G on the closed interval 0<=t<=8 is given by the function -2t/18 * sin(t²/18).