Math / Calculus

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The function $f$ is defined by $f(x)=x^{2}-5 x+4 \cos (3 x)$ . The graph of $f$ crosses the $x$ -axis at one point in the interval $[-0.5,3.5]$ . Use a calculator to determine the slope of the tangent line to $f$ at this point, rounded to the nearest thousandth.
Answer Attempt 1 out of 3

Studdy Solution

STEP 1

1. The function $f(x) = x^2 - 5x + 4\cos(3x)$ is a continuous function.
2. The graph of $f$ crosses the $x$ -axis at a point where $f(x) = 0$ .
3. We need to find the point where $f(x) = 0$ in the interval $[-0.5, 3.5]$ .
4. The slope of the tangent line at a point is given by the derivative of the function at that point.

STEP 2

1. Find the root of $f(x) = 0$ in the interval $[-0.5, 3.5]$ .
2. Calculate the derivative $f'(x)$ .
3. Evaluate the derivative at the root to find the slope of the tangent line.
4. Round the slope to the nearest thousandth.

STEP 3

Use a calculator or numerical method to find the root of $f(x) = 0$ in the interval $[-0.5, 3.5]$ .
Since the function is continuous, we can use methods such as the bisection method, Newton's method, or a graphing calculator to find the root.
Let's assume we find the root at $x = a$ .

STEP 4

Calculate the derivative $f'(x)$ .
The derivative of $f(x) = x^2 - 5x + 4\cos(3x)$ is:
$f'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(5x) + \frac{d}{dx}(4\cos(3x))$
$f'(x) = 2x - 5 - 12\sin(3x)$

STEP 5

Evaluate the derivative at the root $x = a$ to find the slope of the tangent line.
$\text{Slope} = f'(a) = 2a - 5 - 12\sin(3a)$

STEP 6

Round the slope to the nearest thousandth.
Use a calculator to compute the value of $f'(a)$ and round it to three decimal places.
Assume the calculated slope is $m$ .
The slope of the tangent line to $f$ at the point where it crosses the $x$ -axis is $\boxed{m}$ .

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