Math  /  Calculus

Question例 2 求函数 y=x3y=x^{3}x=2,Δx=0.02x=2, \Delta x=0.02 时的微分. 解 先求函数在任意点 xx 的微分 dy=(x3)Δx=3x2Δx\mathrm{d} y=\left(x^{3}\right)^{\prime} \Delta x=3 x^{2} \Delta x

Studdy Solution

STEP 1

1. The function given is y=x3 y = x^3 .
2. We are asked to find the differential of y y when x=2 x = 2 and Δx=0.02 \Delta x = 0.02 .
3. The differential dy \mathrm{d}y can be found using the derivative of y y with respect to x x and then multiplying by Δx \Delta x .

STEP 2

1. Find the derivative of the function y=x3 y = x^3 .
2. Evaluate the derivative at x=2 x = 2 .
3. Compute the differential dy \mathrm{d}y using the evaluated derivative and the given Δx \Delta x .

STEP 3

Find the derivative of the function y=x3 y = x^3 .
dydx=ddx(x3)=3x2 \frac{dy}{dx} = \frac{d}{dx}(x^3) = 3x^2

STEP 4

Evaluate the derivative at x=2 x = 2 .
dydxx=2=3(2)2=12 \left. \frac{dy}{dx} \right|_{x=2} = 3(2)^2 = 12

STEP 5

Compute the differential dy \mathrm{d}y using the evaluated derivative and the given Δx \Delta x .
dy=3x2Δx=120.02=0.24 \mathrm{d}y = 3x^2 \Delta x = 12 \cdot 0.02 = 0.24
The differential dy \mathrm{d}y when x=2 x = 2 and Δx=0.02 \Delta x = 0.02 is 0.24 0.24 .

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