Question

Find the average rate of change for f(t)=2t2+9 over the interval [6,6+h]. Explain your worka) Calculate f(6) and f(6+h).

b) Use the difference quotient: (f(6+h) - f(6))/h.
c) Simplify the expression obtained in step b.
d) The result is the average rate of change over the interval [6,6+h].

Answer 1

To calculate the average rate of change of the function over the interval [6, 6+h], compute f(6) and f(6+h), use the difference quotient, and then simplify the result to obtain the final expression, 24 + 2h.

Step-by-step explanation:

The student is asked to find the average rate of change of the function f(t) = 2t2 + 9 over the interval [6, 6+h].

1. First, calculate f(6), which is f(6) = 2(6)2 + 9 = 2(36) + 9 = 72 + 9 = 81.

2. Next, calculate f(6+h), which is f(6+h) = 2(6+h)2 + 9.

3. Then, use the difference quotient: (f(6+h) - f(6))/h, which gives us ((2(6+h)2 + 9) - 81)/h.

4. Finally, simplify the expression from the previous step to get the average rate of change:
   (2(62 + 12h + h2) + 9 - 81)/h = (2(12h + h2))/h = 24 + 2h. The result is the average rate of change over the given interval.