sections 1.3 - 1.4\nscore: 4/21 answered: 4/21\nquestion 5\nfind the average rate of change of ( f(x)=4…

sections 1.3 - 1.4\nscore: 4/21 answered: 4/21\nquestion 5\nfind the average rate of change of ( f(x)=4 x^{2}-7 ) on the interval ( 5, t ). your answer will be an expression involving ( t )

sections 1.3 - 1.4\nscore: 4/21 answered: 4/21\nquestion 5\nfind the average rate of change of ( f(x)=4 x^{2}-7 ) on the interval ( 5, t ). your answer will be an expression involving ( t )

Answer

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 5), (b=t), and (f(x)=4x^{2}-7).

Step2: Calculate (f(5)) and (f(t))

  • For (x = 5), (f(5)=4\times5^{2}-7=4\times25 - 7=100 - 7=93).
  • For (x=t), (f(t)=4t^{2}-7).

Step3: Substitute into the formula

[ \begin{align*} \frac{f(t)-f(5)}{t - 5}&=\frac{(4t^{2}-7)-93}{t - 5}\ &=\frac{4t^{2}-100}{t - 5}\ &=\frac{4(t^{2}-25)}{t - 5}\ &=\frac{4(t + 5)(t - 5)}{t - 5}\ &=4(t + 5)\ &=4t+20 \end{align*} ]

Answer:

(4t + 20)