91Ó°ÊÓ

Understanding derivatives is a crucial step in working with functions like logarithms. When dealing with the function \( f(x) = \log_{3}(x) \), the derivative indicates the rate at which the function changes at a certain point. For logarithmic functions \( \log_{b}(x) \), the general formula for the derivative is \( \frac{1}{x \ln(b)} \).

This means, for \( f(x) = \log_{3}(x) \), the derivative \( f'(x) \) becomes \( \frac{1}{x \ln(3)} \). The natural logarithm, denoted as \( \ln \), is used here to account for the base change from base \( b \) to the natural logarithm's base \( e \).

At \( x = 3 \), to find how steep the curve is, we substitute \( x = 3 \) into the derivative. Thus, \( f'(3) = \frac{1}{3 \ln(3)} \). By plugging in an approximate value of \( \ln(3) \) as 1.0986, the computation becomes manageable and results in \( 0.303 \) approximately. This value approximates the slope at that specific point on the curve.

Tangent line approximation, often referred to as linearization, helps us approximate the values of functions near a point. This is particularly handy when the function is complicated but behaves almost linearly at small intervals. Linearizing a function involves using its value and its derivative at a specific point to form a simple linear function.

The formula for linearization \( L(x) \) at a point \( x = a \) is derived as: \[ L(x) = f(a) + f'(a)(x - a) \]
In this case, for \( f(x) = \log_{3}(x) \) and \( a = 3 \), we've calculated \( f(3) = 1 \) and \( f'(3) = 0.303 \). By plugging these into our formula, we get:
\[ L(x) = 1 + 0.303(x - 3) \]
We round the coefficient \( 0.303 \) to \( 0.30 \) for a nicely simplified result.

Thus, our tangent line approximation at \( x = 3 \) is \( L(x) = 1 + 0.30(x - 3) \), which makes it easier to compute values close to \( x = 3 \). This linear function closely matches the logarithmic curve at this point and gives a straight-line approximation.

Graphing the function \( f(x) = \log_{3}(x) \) alongside its linear approximation presents a clear picture of how linearization works.
In our specific scenario, we plot both the function and its linearization \( L(x) = 1 + 0.30(x - 3) \) in the range \( 0 \leq x \leq 8 \) and \( 2 \leq x \leq 4 \).

This range covers a portion of the curve where the approximation holds true. For effective visualization:

• The logarithmic function shows a gently increasing curve since logarithms increase slowly as x grows.
• The tangent line, meanwhile, is straight, capturing the behavior accurately near \( x = 3 \).

By using graphing tools, one can observe that near \( x = 3 \), the tangent line and the curve are nearly indistinguishable. As you move further from this point, the difference becomes noticeable, showcasing the limitation of linear approximations.
Concluding, graphing not only aids in understanding function behavior at specific points but also displays the practicality and bounds of linearization.