Math Snap

STEP 1

What is this asking?
We need to find a formula for the tiny change in $y$ when there's a tiny change in $x$, given that $y = x \cos(x)$.
Then, we need to calculate this tiny change in $y$ when $x$ is 4.71239 and the tiny change in $x$ is 0.3.
Watch out!
Remember the product rule for derivatives!
Also, make sure your calculator is in radians mode when evaluating the cosine function.

STEP 2

1. Find the Differential
2. Evaluate at Given Values

STEP 3

We are given the function $y = x \cos(x)$.

STEP 4

To find the differential, we need to find the derivative of $y$ with respect to $x$.
Since $y$ is a product of two functions, $x$ and $\cos(x)$, we'll use the product rule:
$$\frac{dy}{dx} = \frac{d}{dx}(x) \cdot \cos(x) + x \cdot \frac{d}{dx}(\cos(x)).$$

STEP 5

The derivative of $x$ with respect to $x$ is simply 1.
The derivative of $\cos(x)$ with respect to $x$ is $- \sin(x)$.
So,
$$\frac{dy}{dx} = 1 \cdot \cos(x) + x \cdot (-\sin(x)) = \cos(x) - x \sin(x).$$

STEP 6

The differential $dy$ is given by:
$$dy = \frac{dy}{dx} dx = (\cos(x) - x \sin(x)) dx.$$

STEP 7

We are given $x_0 = 4.71239$ and $dx = 0.3$.
Let's plug these values into our differential:
$$dy = (\cos(4.71239) - 4.71239 \cdot \sin(4.71239)) \cdot 0.3.$$

STEP 8

Using a calculator in radians mode:
$\cos(4.71239) \approx 0$, and $\sin(4.71239) \approx -1$.

STEP 9

Substituting these values back into the equation:
$$dy \approx (0 - 4.71239 \cdot (-1)) \cdot 0.3 = (4.71239) \cdot 0.3 \approx 1.413717.$$ Rounding to 4 decimal places, we get $dy \approx 1.4137$.

The differential of $y$ is $dy = (\cos(x) - x \sin(x)) dx$.
At $x_0 = 4.71239$ and $dx = 0.3$, the differential is approximately $1.4137$.