Problem 42 - Calculus - AP Edition

Question

Find the equation of the tangent line to the graph of \(y=f(x)\) at \(x=x_{0}\). $$f(x)=\log x ; x_{0}=10$$

Step-by-Step Solution

Answer

The equation of the tangent line is \(y = \frac{1}{10}x\).

Step 1: Find the Derivative

To find the equation of the tangent line, we first need the derivative of the function, which represents the slope of the tangent line at a given point. The function given is \(f(x) = \log x\). The derivative of this function with respect to \(x\) is \(f'(x) = \frac{1}{x}\).

Step 2: Evaluate the Derivative at \(x_0\)

Now, we evaluate the derivative at the point \(x_0 = 10\). This gives us the slope of the tangent line at the point. \(f'(10) = \frac{1}{10}\). So, the slope of the tangent line is \(\frac{1}{10}\).

Step 3: Find the y-coordinate at \(x_0\)

Next, we need the y-coordinate of the point of tangency. We do this by evaluating the original function \(f(x)\) at \(x_0 = 10\). So, \(f(10) = \log 10\). Since \(\log 10 = 1\) (using base \(10\)), the y-coordinate is \(1\).

Step 4: Write the Equation of the Tangent Line

We use the point-slope form of the equation of a line, which is \(y-y_1 = m(x-x_1)\), where \((x_1, y_1)\) is the point of tangency and \(m\) is the slope. From steps 2 and 3, we know the point of tangency is \((10, 1)\) and the slope \(m = \frac{1}{10}\). Plugging these into the point-slope formula gives: \[y - 1 = \frac{1}{10}(x - 10)\].

Step 5: Simplify the Equation

Finally, we simplify the equation from Step 4 to get the tangent line in slope-intercept form. Simplifying \[y - 1 = \frac{1}{10}(x - 10)\] gives \[y - 1 = \frac{1}{10}x - 1\]. Adding 1 to both sides results in \[y = \frac{1}{10}x\].

Key Concepts

Derivative Calculus

In calculus, the derivative of a function at any given point helps us understand the rate at which the function is changing at that point. To find the equation of the tangent line to a curve at a specific point, we first need to determine the slope of this tangent line. This slope is nothing but the value of the derivative at that particular point.

If the function is given as \( f(x) = \log x \), the process of differentiation involves finding its derivative with respect to \( x \). The derivative of \( \log x \) is \( f'(x) = \frac{1}{x} \). This formula tells us how quickly the logarithmic function changes for small changes in \( x \).

When given a specific point \( x_{0} \), in this case \( x_{0} = 10 \), we evaluate the derivative at that point, \( f'(10) = \frac{1}{10} \). This results in a slope of \( \frac{1}{10} \), which is crucial for writing the equation of the tangent line.

Logarithmic Functions

Logarithmic functions are used frequently in calculus and other mathematical contexts due to their unique properties. A logarithm is the inverse operation to exponentiation and helps in solving equations where the variable is an exponent.

The function \( f(x) = \log x \) represents a logarithm base 10, which is often found in real-world applications including scientific data analysis and sound intensity. At \( x = 10 \), the logarithmic function evaluates to 1 since \( \log 10 = 1 \). This point \( (10, 1) \) is significant in our problem because it provides the \( y \)-coordinate in the equation of the tangent line.

• Characteristics of logarithmic functions:
• They grow more slowly than linear or polynomial functions.
• They are defined only for positive values of \( x \).
• Their graph is a smooth curve that passes through the points \( (1, 0) \) and \( (10, 1) \).

Point-Slope Form

The point-slope form of a linear equation is especially useful in calculus for writing the equation of a tangent line. It is expressed as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

In our example, we know the slope \( m = \frac{1}{10} \) and the point of tangency is \( (10, 1) \). Plugging these into the point-slope formula gives \( y - 1 = \frac{1}{10}(x - 10) \). This formula precisely defines the tangent line to the graph of \( y = \log x \) at the point \( x = 10 \).

Finally, simplifying this equation can often involve rewriting it in slope-intercept form, \( y = mx + b \), where \( b \) is the y-intercept. Here, simplifying gives us \( y = \frac{1}{10}x \), which confirms the straight line equation of the tangent.