Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Estimate the slope of the tangent to the graph of $$y=\log _{10} x$$ at the point \(\left(3, \log _{10} 3\right)\) correct to three decimal digits.

Step-by-Step Solution

Answer

The slope of the tangent at the point is approximately 0.145.

Step 1: Differentiate the Function

To find the slope of the tangent to the graph, we need to differentiate the function. The given function is \( y = \log_{10} x \). The derivative of \( \log_{10} x \) with respect to \( x \) can be found using the rule for differentiating logarithmic functions:\[ \frac{d}{dx}(\log_{10} x) = \frac{1}{x \ln(10)} \]This gives us the derivative \( y' = \frac{1}{x \ln(10)} \).

Step 2: Evaluate the Derivative at the Point

Now, we need to find the slope of the tangent at the specific point \( (3, \log_{10} 3) \). This requires evaluating the derivative at \( x = 3 \):\[ y'(3) = \frac{1}{3 \ln(10)} \]Compute \( \ln(10) \) using a calculator to get approximately \( 2.302 \). So, the expression becomes:\[ y'(3) = \frac{1}{3 \times 2.302} \]

Step 3: Compute the Numerical Value

Simplify the expression: \[ y'(3) = \frac{1}{6.906} \approx 0.145 \]Thus, the slope of the tangent to the graph at the given point is approximately 0.145.

Key Concepts

Derivative of logarithmic functions

Understanding logarithmic functions and their derivatives is crucial in calculus. The logarithmic function given in the problem is \( y = \log_{10} x \). To find its derivative, we employ the rule for differentiating logarithmic functions, particularly for base 10 logs. The formula is:

• \( \frac{d}{dx}(\log_{10} x) = \frac{1}{x \ln(10)} \)

This derivative expression tells us how the function \( \log_{10} x \) changes for small changes in \( x \). If you're dealing with natural logarithms (base \( e \), \( \ln x \)), the derivative takes a simpler form: \( \frac{d}{dx}(\ln x) = \frac{1}{x} \).
For base 10, however, it's necessary to include the natural logarithm of 10 in the denominator of our expression. This additional factor comes from the change of base formula in logarithms. So, the derivative simplifies to \( y' = \frac{1}{x \ln(10)} \), which is the slope of the tangent line at any point \( x \) on the graph.

Tangent to a curve

The concept of a tangent line to a curve is a fundamental idea in calculus. A tangent line is a straight line that just "touches" the curve at a specific point without crossing it. This line represents the instantaneous rate of change of the function at that point. To find this line, we need to identify its slope, which is determined by differentiating the function.
In the case of our problem, the slope of the tangent to the curve \( y = \log_{10} x \) at the point \( (3, \log_{10} 3) \) is found by evaluating the derivative at \( x=3 \). The derivative gives us:

• \( y'(3) = \frac{1}{3 \times \ln(10)} \)

Substituting the approximate value of \( \ln(10) \) (which is 2.302) into the expression, we determine that the slope is approximately 0.145. This means that at \( x=3 \), the curve behaves linearly and increases by about 0.145 units vertically for each unit increase horizontally. Understanding this concept allows us to interpret how functions change at very specific points.

Differential calculus

Differential calculus is all about studying change. It's concerned with how functions increase or decrease, how their curves look, and what their rates of change are at any given point. The derivative, which you compute in differential calculus, offers insight into the slope or rate of change of a function at different points on its curve.
In practical terms:

• It helps find the slope of the tangent line to the curve.
• It determines how the function behaves locally around a particular point.

From our problem, by differentiating \( y=\log_{10} x \) and evaluating it at \( x=3 \), we see a clear application of differential calculus. It provides a precise slope, 0.145, informing us how steep the curve is at that point.
More broadly, differential calculus tools like derivatives enable scientists, engineers, and economists to model real-world problems, predict trends, and make informed decisions based on the rates of change they observe. It's a gateway to understanding the dynamic nature of systems and functions in numerous fields.