A function y = 5x2 + 10x is defined over an open interval x = (1, 2). At least at one point in this interval, \(\frac{dy}{dx}\) is exactly
25
Understanding the Function and Interval
We are given a function \(y = 5x^2 + 10x\). This function is defined over an open interval for \(x\), specifically \(x \in (1, 2)\). An open interval means that the values \(x=1\) and \(x=2\) are not included in the interval, so we consider values of \(x\) strictly between 1 and 2, i.e., \(1 < x < 2\).
Calculating the Derivative
The question asks about the value of the derivative, \(\frac{dy}{dx}\), at a point within this interval. First, we need to find the derivative of the function \(y\) with respect to \(x\). We use the power rule for differentiation, which states that \(\frac{d}{dx}(ax^n) = anx^{n-1}\).
Applying this rule to our function:
$$ \frac{dy}{dx} = \frac{d}{dx}(5x^2 + 10x) $$
Separating the terms:
$$ \frac{dy}{dx} = \frac{d}{dx}(5x^2) + \frac{d}{dx}(10x) $$
Using the power rule:
$$ \frac{dy}{dx} = (5 \cdot 2)x^{(2-1)} + (10 \cdot 1)x^{(1-1)} $$
$$ \frac{dy}{dx} = 10x^1 + 10x^0 $$
Since \(x^1 = x\) and \(x^0 = 1\), the derivative is:
$$ \frac{dy}{dx} = 10x + 10 $$
Evaluating the Derivative in the Interval
Now we need to determine the range of values for \(\frac{dy}{dx}\) when \(x\) is in the interval \((1, 2)\). Since \(\frac{dy}{dx} = 10x + 10\) is a linear function with a positive slope (10), it is an increasing function. This means as \(x\) increases, \(\frac{dy}{dx}\) also increases.
Let's find the values of \(\frac{dy}{dx}\) at the boundaries of the interval (even though the interval is open, these help define the range):
- At \(x = 1\): \(\frac{dy}{dx} = 10(1) + 10 = 20\)
- At \(x = 2\): \(\frac{dy}{dx} = 10(2) + 10 = 30\)
Since \(x\) is strictly between 1 and 2 (\(1 < x < 2\)), the value of the derivative \(\frac{dy}{dx}\) will be strictly between 20 and 30 (\(20 < \frac{dy}{dx} < 30\)).
Finding the Exact Derivative Value
The question states that \(\frac{dy}{dx}\) is *exactly* a certain value at *at least one point* in the interval \((1, 2)\). We need to find which of the given options falls within the range \((20, 30)\).
The options are 20, 25, 30, and 35.
- 20: This value is the derivative at \(x=1\), but \(x=1\) is not included in the open interval \((1, 2)\).
- 25: This value lies within the range \((20, 30)\). Let's check if there's an \(x\) in \((1, 2)\) that gives this value:
Set \(\frac{dy}{dx} = 25\):
$$ 10x + 10 = 25 $$
Subtract 10 from both sides:
$$ 10x = 15 $$
Divide by 10:
$$ x = \frac{15}{10} = 1.5 $$
Since \(1.5\) is indeed within the open interval \((1, 2)\), the derivative is exactly 25 at \(x = 1.5\).
- 30: This value is the derivative at \(x=2\), but \(x=2\) is not included in the open interval \((1, 2)\).
- 35: This value is greater than 30, so it is outside the range of possible derivative values for \(x \in (1, 2)\).
Therefore, at least at one point in the interval \((1, 2)\), the derivative \(\frac{dy}{dx}\) is exactly 25.
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