F(x) =5x^2. Find a value A such that the average rate of change of f(x) from 1 to A equals 50.

Solution:
1. Given:
- Function: f(x) = 5x^2
- Average rate of change from x=1 to x=A equals 50.

2. The formula for the average rate of change is:
\text{Average rate of change} = \frac{f(A) - f(1)}{A - 1}

3. Calculate f(1):
f(1) = 5(1)^2 = 5

4. Substitute into the average rate of change formula:
\frac{f(A) - 5}{A - 1} = 50

5. Substitute f(A) = 5A^2 into the equation:
\frac{5A^2 - 5}{A - 1} = 50

6. Multiply both sides by (A - 1):
5A^2 - 5 = 50(A - 1)

7. Distribute and simplify:
5A^2 - 5 = 50A - 50
5A^2 - 50A + 45 = 0

8. Divide by 5:
A^2 - 10A + 9 = 0

9. Factor the quadratic equation:
(A - 9)(A - 1) = 0

10. Solve for A:
- A - 9 = 0 or A - 1 = 0
- A = 9 or A = 1

11. Since A \neq 1 (because we are finding the change from x = 1 to x = A:
- The valid solution is A = 9.

So, the value of A is:
A = 9