Math / Calculus

QuestionRates of Change Assignment
1. Determine the average rate of change of $y$ in the function $y=2 x^{3}+7 x^{2}+2 x-3$ over the interval [3, 5]. [5 marks]

Studdy Solution

STEP 1

1. The function given is $y = 2x^3 + 7x^2 + 2x - 3$ .
2. The average rate of change of a function over an interval $[a, b]$ is given by the formula: $\frac{y(b) - y(a)}{b - a}$ .
3. The interval for this problem is $[3, 5]$ .

STEP 2

1. Evaluate the function $y$ at the endpoints of the interval, $x = 3$ and $x = 5$ .
2. Use the values obtained to calculate the average rate of change using the formula $\frac{y(5) - y(3)}{5 - 3}$ .

STEP 3

Evaluate the function $y$ at $x = 3$ .
$y(3) = 2(3)^3 + 7(3)^2 + 2(3) - 3$
Simplify the expression to find $y(3)$ .
$y(3) = 2(27) + 7(9) + 6 - 3$ $y(3) = 54 + 63 + 6 - 3$ $y(3) = 120$

STEP 4

Evaluate the function $y$ at $x = 5$ .
$y(5) = 2(5)^3 + 7(5)^2 + 2(5) - 3$
Simplify the expression to find $y(5)$ .
$y(5) = 2(125) + 7(25) + 10 - 3$ $y(5) = 250 + 175 + 10 - 3$ $y(5) = 432$

STEP 5

Calculate the average rate of change of $y$ over the interval $[3, 5]$ using the formula $\frac{y(5) - y(3)}{5 - 3}$ .
$\text{Average Rate of Change} = \frac{y(5) - y(3)}{5 - 3} = \frac{432 - 120}{5 - 3}$ $\text{Average Rate of Change} = \frac{312}{2} = 156$
In conclusion, the average rate of change of $y$ in the function $y = 2x^3 + 7x^2 + 2x - 3$ over the interval $[3, 5]$ is:
$156$

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