Question Statement

Approximate the value of $\tan \left(\frac{\pi }{4}+0.1\right)$ using differentials.

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Background and Explanation

This problem uses linear approximation (or differential approximation), which estimates the value of a function near a known point using the tangent line. When $\Delta x$ is small, $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\Delta x$.

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Solution

We want to approximate $\tan \left(\frac{\pi }{4}+0.1\right)$ using the method of differentials.

Step 1: Choose the function and values

Let $f(x)=\tan x$

Choose a base point $x=\frac{\pi }{4}$ (since $\tan \frac{\pi }{4}$ is known exactly) and the increment $\Delta x=dx=0.1$.

Note that $\frac{\pi }{4}+0.1=x+\Delta x$, so we can apply the approximation formula.

Step 2: Find the derivative

Differentiate $f(x)=\tan x$ with respect to $x$:

$f^{\prime }(x)={\sec }^{2}x$

Step 3: Apply the linear approximation formula

The formula for differential approximation is:

$f(x+\Delta x)\approx f(x)+f^{\prime }(x)dx$

Substituting our specific function:

$\tan (x+\Delta x)\approx \tan x+{\sec }^{2}x\cdot dx$

Step 4: Substitute the numerical values

Plugging in $x=\frac{\pi }{4}$ and $dx=0.1$:

$\tan \left(\frac{\pi }{4}+0.1\right)\approx \tan \frac{\pi }{4}+{\sec }^2\frac{\pi }{4}\cdot (0.1)$

Step 5: Evaluate the trigonometric functions

We know that:

• $\tan \frac{\pi }{4}=1$
• $\sec \frac{\pi }{4}=\frac{1}{\cos \frac{\pi }{4}}=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2}$

Therefore:

• ${\sec }^2\frac{\pi }{4}=(\sqrt{2}{)}^2=2$

Step 6: Calculate the final approximation

$\begin{array}{rl}\tan \left(\frac{\pi }{4}+0.1\right)& \approx 1+(2)(0.1)\\ & =1+0.2\\ & =1.2\end{array}$

Hence, $\tan \left(\frac{\pi }{4}+0.1\right)\approx 1.2$.

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Key Formulas or Methods Used

• Linear Approximation Formula: $f(x+\Delta x)\approx f(x)+f^{\prime }(x)dx$
• Derivative of Tangent: $\frac{d}{dx}(\tan x)={\sec }^{2}x$
• Trigonometric Values: $\tan \frac{\pi }{4}=1$ and $\sec \frac{\pi }{4}=\sqrt{2}$

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Summary of Steps

1. Identify $f(x)=\tan x$ and choose $x=\frac{\pi }{4}$ (where the value is known) with $\Delta x=0.1$
2. Compute the derivative $f^{\prime }(x)={\sec }^{2}x$
3. Apply the linear approximation formula: $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\Delta x$
4. Substitute values: $\tan \frac{\pi }{4}+{\sec }^2\frac{\pi }{4}\cdot (0.1)$
5. Evaluate: $1+2(0.1)=1.2$