Question

What is the average rate of change of the function f(x)=20(14)xf(x)=20(14)x from x = 1 to x = 2? Enter your answer, as a decimal, in the box. Do not round your answer.

Answer 1

$\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\ -------------------------------\\\\ f(x)= 20(14)^x \qquad \begin{cases} x_1=1\\ x_2=2 \end{cases}\implies \cfrac{f(2)-f(1)}{2-1} \\\\\\ \cfrac{[20(14)^2]~~-~~[20(14)^1]}{1}\implies \cfrac{20\cdot 196~~-~~20\cdot 14}{1} \\\\\\ \cfrac{3920~~-~~280}{1}\implies \cfrac{3640}{1}\implies 3640$

Answer 2

Answer:

Average rate of change(A(x)) of f(x) over a interval [a,b] is given by:

$A(x) = \frac{f(b)-f(a)}{b-a}$

Given the function:

$f(x) = 20 \cdot(\frac{1}{4})^x$

We have to find the average rate of change from x = 1 to x= 2

At x = 1

then;

$f(x) = 20 \cdot(\frac{1}{4})^1 = 5$

At x = 2

then;

$f(x) = 20 \cdot(\frac{1}{4})^2=20 \cdot \frac{1}{16} = 1.25$

Substitute these in above formula we have;

$A(x) = \frac{f(2)-f(1)}{2-1}$

⇒$A(x) = \frac{1.25-5}{1}=-3.75$

therefore, average rate of change of the function f(x) from x = 1 to x = 2 is, -3.75