Math / Calculus

QuestionThe Differential of a Function.
Find the differential of the given function. Then, evaluate the differential at the indicated values.
If $y=x \cos (x)$ then the differential of $y$ is $d y=\square$
Note: Type $d x$ for the differential of $x$ Evaluate the differential of $y$ at $x_{0}=4.71239, d x=0.3$ $\left.d y\right|_{\substack{x=x_{0} \\ d x=0.3}}=\square$
Note: Enter your answer accurate to 4 decimal places if it is not an Integrer.

Studdy Solution

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 $dy \approx 1.4137$ .

STEP 10

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 $1.4137$ .

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Studdy Solution

STEP 1

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 $dy \approx 1.4137$ .

STEP 10

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 $1.4137$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the Differential2. Evaluate at Given Values

STEP 3

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

STEP 4

STEP 5

STEP 6

The differential dydydy is given by: dy=dydxdx=(cos⁡(x)−xsin⁡(x))dx. dy = \frac{dy}{dx} dx = (\cos(x) - x \sin(x)) dx. dy=dxdy​dx=(cos(x)−xsin(x))dx.

STEP 7

We are given x0=4.71239x_0 = 4.71239x0​=4.71239 and dx=0.3dx = 0.3dx=0.3.Let's plug these values into our differential: dy=(cos⁡(4.71239)−4.71239⋅sin⁡(4.71239))⋅0.3. dy = (\cos(4.71239) - 4.71239 \cdot \sin(4.71239)) \cdot 0.3. dy=(cos(4.71239)−4.71239⋅sin(4.71239))⋅0.3.

STEP 8

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

STEP 9

STEP 10

The differential of yyy is dy=(cos⁡(x)−xsin⁡(x))dxdy = (\cos(x) - x \sin(x)) dxdy=(cos(x)−xsin(x))dx. At x0=4.71239x_0 = 4.71239x0​=4.71239 and dx=0.3dx = 0.3dx=0.3, the differential is approximately 1.41371.41371.4137.

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1. Find the Differential
2. Evaluate at Given Values

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

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

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

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

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 $1.4137$ .