3.1 Q-3

Question Statement

Use differentials to approximate the following values:

1. $\sqrt[4]{17}$
2. $(31)^{\frac{1}{5}}$
3. $\cos(29^\circ)$
4. $\sin(61^\circ)$

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Background and Explanation

Differentials are used to approximate the values of functions near a given point. The key idea is that for a small change in $x$ (denoted $\delta x$ or $dx$), the change in $y$ (denoted $\delta y$ or $dy$) can be approximated as $dy = f'(x) dx$. This approach relies on linearization, where the tangent line approximates the curve near a point.

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Solution

Part 1: Approximation of $\sqrt[4]{17}$

1. Let $y = x^{1/4}$.

   • Choose $x = 16$ (a perfect fourth power near 17).
   • $\delta x = 17 - 16 = 1$.

2. Derivative:

$$\frac{dy}{dx} = \frac{1}{4} x^{-3/4}.$$

3. Compute $dy$:

$$dy = \frac{1}{4} (16)^{-3/4} \cdot 1 = \frac{1}{4 \cdot 8} = \frac{1}{32} = 0.03125.$$

4. Approximation:

$$\sqrt[4]{17} \approx \sqrt[4]{16} + dy = 2 + 0.03125 = 2.03125.$$

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Part 2: Approximation of $(31)^{\frac{1}{5}}$

1. Let $y = x^{1/5}$.

   • Choose $x = 32$ (a perfect fifth power near 31).
   • $\delta x = 31 - 32 = -1$.

2. Derivative:

$$\frac{dy}{dx} = \frac{1}{5} x^{-4/5}.$$

3. Compute $dy$:

$$dy = \frac{1}{5} (32)^{-4/5} \cdot (-1) = \frac{-1}{5 \cdot 16} = \frac{-1}{80} = -0.0125.$$

4. Approximation:

$$(31)^{\frac{1}{5}} \approx (32)^{\frac{1}{5}} + dy = 2 - 0.0125 = 1.9875.$$

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Part 3: Approximation of $\cos(29^\circ)$

1. Let $y = \cos x$.

   • Choose $x = 30^\circ = \frac{\pi}{6}$.
   • $\delta x = 29^\circ - 30^\circ = -1^\circ = -\frac{\pi}{180}$ radians.

2. Derivative:

$$\frac{dy}{dx} = -\sin x.$$

3. Compute $dy$:

$$dy = (-\sin(30^\circ))(-\frac{\pi}{180}) = \sin(30^\circ) \cdot \frac{\pi}{180} = \frac{1}{2} \cdot \frac{\pi}{180} = 0.0087.$$

4. Approximation:

$$\cos(29^\circ) \approx \cos(30^\circ) + dy = 0.866 + 0.0087 = 0.8747.$$

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Part 4: Approximation of $\sin(61^\circ)$

1. Let $y = \sin x$.

   • Choose $x = 60^\circ = \frac{\pi}{3}$.
   • $\delta x = 61^\circ - 60^\circ = 1^\circ = \frac{\pi}{180}$ radians.

2. Derivative:

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

3. Compute $dy$:

$$dy = \cos(60^\circ) \cdot \frac{\pi}{180} = \frac{1}{2} \cdot \frac{\pi}{180} = 0.0087.$$

4. Approximation:

$$\sin(61^\circ) \approx \sin(60^\circ) + dy = 0.866 + 0.0087 = 0.8747.$$

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Key Formulas or Methods Used

1. Linear approximation: $dy = f'(x) dx$.
2. Derivatives:
   • $\frac{d}{dx}(x^n) = n x^{n-1}$.
   • $\frac{d}{dx}(\sin x) = \cos x$.
   • $\frac{d}{dx}(\cos x) = -\sin x$.

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Summary of Steps

1. Identify a nearby point where the function is easily computable.
2. Calculate the derivative of the function.
3. Use $dy = f'(x) dx$ to estimate the change.
4. Add $dy$ to the value of the function at the nearby point to approximate the desired value.

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