3.1 Q-1

Question Statement

Find the change in $y$ ($\delta y$) and the differential ($dy$) in the following cases:

1. $y = x^2 - 1$ when $x$ changes from 3 to 3.02.
2. $y = x^2 + 2x$ when $x$ changes from 2 to 1.8.
3. $y = \sqrt{x}$ when $x$ changes from 4 to 4.41.

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Background and Explanation

To calculate $\delta y$, we measure the actual change in $y$ as $x$ changes by a small amount $\delta x$. It involves substituting $x + \delta x$ into the given equation of $y$.

To calculate $dy$, we use the concept of differentials, applying calculus to find the derivative of $y$ with respect to $x$, then multiplying it by $dx$, where $dx = \delta x$.

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Solution

Case 1: $y = x^2 - 1$, $x$ changes from 3 to 3.02

Step-by-Step:

1. Initial values:
   $x = 3$, $\delta x = dx = 3.02 - 3 = 0.02$.

2. Calculate $y$ at $x = 3$:

$$y = x^2 - 1 = 3^2 - 1 = 8.$$

3. Calculate $\delta y$:
   Substitute $x + \delta x = 3.02$ into the equation:

$$y + \delta y = (x + \delta x)^2 - 1 = (3.02)^2 - 1.$$ $$y + \delta y = 9.1204 - 1 = 8.1204.$$ $$\delta y = 8.1204 - 8 = 0.1204.$$

4. Calculate $dy$:
   The derivative of $y = x^2 - 1$ is:

$$\frac{dy}{dx} = 2x.$$

Substitute $x = 3$ and $dx = 0.02$:

$$dy = 2x \cdot dx = 2(3)(0.02) = 0.12.$$

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Case 2: $y = x^2 + 2x$, $x$ changes from 2 to 1.8

Step-by-Step:

1. Initial values:
   $x = 2$, $\delta x = dx = 1.8 - 2 = -0.2$.

2. Calculate $y$ at $x = 2$:

$$y = x^2 + 2x = 2^2 + 2(2) = 4 + 4 = 8.$$

3. Calculate $\delta y$:
   Substitute $x + \delta x = 1.8$ into the equation:

$$y + \delta y = (x + \delta x)^2 + 2(x + \delta x).$$ $$y + \delta y = (1.8)^2 + 2(1.8) = 3.24 + 3.6 = 6.84.$$ $$\delta y = 6.84 - 8 = -1.16.$$

4. Calculate $dy$:
   The derivative of $y = x^2 + 2x$ is:

$$\frac{dy}{dx} = 2x + 2.$$

Substitute $x = 2$ and $dx = -0.2$:

$$dy = (2x + 2) \cdot dx = (2(2) + 2)(-0.2) = 6(-0.2) = -1.2.$$

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Case 3: $y = \sqrt{x}$, $x$ changes from 4 to 4.41

Step-by-Step:

1. Initial values:
   $x = 4$, $\delta x = dx = 4.41 - 4 = 0.41$.

2. Calculate $y$ at $x = 4$:

$$y = \sqrt{x} = \sqrt{4} = 2.$$

3. Calculate $\delta y$:
   Substitute $x + \delta x = 4.41$ into the equation:

$$y + \delta y = \sqrt{x + \delta x} = \sqrt{4.41}.$$ $$y + \delta y = 2.1.$$ $$\delta y = 2.1 - 2 = 0.1.$$

4. Calculate $dy$:
   The derivative of $y = \sqrt{x}$ is:

$$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}.$$

Substitute $x = 4$ and $dx = 0.41$:

$$dy = \frac{1}{2\sqrt{x}} \cdot dx = \frac{1}{2\sqrt{4}} \cdot 0.41 = \frac{1}{4} \cdot 0.41 = 0.1025.$$

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Key Formulas or Methods Used

1. $\delta y = f(x + \delta x) - f(x)$: Actual change in $y$.
2. $dy = \frac{dy}{dx} \cdot dx$: Differential approximation of change in $y$.
3. Derivatives:
   • $\frac{d}{dx}(x^2) = 2x$
   • $\frac{d}{dx}(x^2 + 2x) = 2x + 2$
   • $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$

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Summary of Steps

1. Calculate $\delta x$ from the given $x$ values.
2. Find the original $y$ using the given equation.
3. Substitute $x + \delta x$ into the equation to compute $\delta y$.
4. Use derivatives to calculate $dy$ as an approximation for $\delta y$.
5. Compare results of $\delta y$ and $dy$.

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