question\nfind the average rate of change of the function\nf(x), given below, from x = 1 to x = t.\n$$ f ( x…

question\nfind the average rate of change of the function\nf(x), given below, from x = 1 to x = t.\n$$ f ( x ) = - 2 x ^ { 2 } - 3 x - 2 $$\ngive your answer in terms of t.\nprovide your answer below:

question\nfind the average rate of change of the function\nf(x), given below, from x = 1 to x = t.\n$$ f ( x ) = - 2 x ^ { 2 } - 3 x - 2 $$\ngive your answer in terms of t.\nprovide your answer below:

Answer

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) from (x=a) to (x = b) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 1) and (b=t).

Step2: Calculate (f(1)) and (f(t))

  • For (f(1)): Substitute (x = 1) into (f(x)=-2x^{2}-3x - 2). (f(1)=-2(1)^{2}-3(1)-2=-2 - 3-2=-7).
  • For (f(t)): Substitute (x = t) into (f(x)=-2x^{2}-3x - 2). (f(t)=-2t^{2}-3t - 2).

Step3: Substitute into the average - rate - of - change formula

(\frac{f(t)-f(1)}{t - 1}=\frac{(-2t^{2}-3t - 2)-(-7)}{t - 1}). Simplify the numerator: ((-2t^{2}-3t - 2 + 7)=-2t^{2}-3t + 5). Factor the numerator: (-2t^{2}-3t + 5=-(2t^{2}+3t - 5)=-(2t + 5)(t - 1)). So, (\frac{-(2t + 5)(t - 1)}{t - 1}), and since (t\neq1) (denominator cannot be zero), we can cancel out the ((t - 1)) terms.

Answer:

(-2t-5)