Let f(x)=7x2.
Find a value A such that the average rate of change of f(x) from 1 to A equals 84.
A=
Its F(x)= 7x^(2). Sorry.
f(1) = 7(1)^2 = 7
f(A) = 7A^2
avg rate of change = (7a^2 - 7)/(a-1)
(7a^2 - 7)/(a-1) = 84
7a^2 - 7 = 84a - 84
7a^2 - 84a + 77 = 0
a^2 - 12a + 11 = 0
(a-11)(a-1) = 0
a = 11 or a = 1, but we have to reject a = 1 or else we would be dividing by zero
A =11
check:
f(11) = 7(121) = 847
f(1) = 7
rate of change = (847-7)/(11-1)
= 840/10 = 84
looks ok
Thank you very much Reiny. You broke it down perfectly for me.
or, picking up at
(7a^2 - 7)/(a-1) = 84
7(a-1)(a+1) / (a-1) = 84
7(a+1) = 84
a+1 = 12
a = 11
To find the value of A such that the average rate of change of f(x) from 1 to A equals 84, we need to use the formula for average rate of change. The average rate of change of a function f(x) from x = a to x = b is given by the formula:
average rate of change = (f(b) - f(a)) / (b - a)
In this case, we have f(x) = 7x^2 and a = 1. So, we need to solve the equation:
84 = (f(A) - f(1)) / (A - 1)
Let's calculate the values of f(A) and f(1):
f(A) = 7A^2
f(1) = 7(1)^2 = 7
Substituting these values into the equation:
84 = (7A^2 - 7) / (A - 1)
To simplify the equation, let's multiply both sides by (A - 1) to eliminate the denominator:
84(A - 1) = 7A^2 - 7
Open up the brackets:
84A - 84 = 7A^2 - 7
Rearrange the equation to obtain a quadratic equation:
7A^2 - 84A + 77 = 0
Next, we can solve this quadratic equation by factoring or using the quadratic formula. Factoring might not be possible in this case, so let's use the quadratic formula:
A = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, a = 7, b = -84, and c = 77. Plugging these values into the quadratic formula:
A = (-(-84) ± √((-84)^2 - 4(7)(77))) / (2(7))
Simplifying further:
A = (84 ± √(7056 - 2156)) / 14
A = (84 ± √4900) / 14
A = (84 ± 70) / 14
This gives us two possible solutions for A:
1. A = (84 + 70) / 14 = 154 / 14 = 11
2. A = (84 - 70) / 14 = 14 / 14 = 1
Therefore, the two possible values for A such that the average rate of change of f(x) from 1 to A equals 84 are A = 11 and A = 1.