Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=\frac{3}{2 x}\) at \(\left(1, \frac{3}{2}\right)\)

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of \(f(x) = \frac{3}{2x}\) at the point \((1, \frac{3}{2})\) is \(-\frac{3}{4}\), and the equation of the tangent line is \(y = -\frac{3}{4}x + \frac{5}{2}\).

Step 1: Find the derivative of the function.

The given function is \(f(x) = \frac{3}{2x}\). To find the derivative with respect to \(x\), we will apply the power rule: \(f(x) = 3\cdot (2x)^{-1}\) Now use the power rule, \((x^n)' = nx^{n-1}\): \(f'(x) = 3(-1)(2x)^{-2}\) Simplify the expression: \(f'(x) = -\frac{3}{4x^2}\)

Step 2: Find the slope at the given point.

We have found the derivative \(f'(x) = -\frac{3}{4x^2}\), which gives the slope of the tangent line for any point on the graph of \(f(x)\). Now, we need to evaluate the derivative at the given point \((1, \frac{3}{2})\). Since the point lies on the graph of the function, we will plug in the given \(x\)-coordinate, \(1\), into the derivative: \(m = f'(1) = -\frac{3}{4(1^2)}\) \(m = -\frac{3}{4}\) So, the slope of the tangent line at \((1, \frac{3}{2})\) is \(-\frac{3}{4}\).

Step 3: Determine the equation of the tangent line.

Now that we have the slope of the tangent line (\(m = -\frac{3}{4}\)) and the point \((1,\frac{3}{2})\), we can use the point-slope form of the line to find the equation of the tangent line: \(y - y_1 = m(x - x_1)\) Plugging in the slope and the given point: \(y - \frac{3}{2} = -\frac{3}{4}(x - 1)\) Now, solve for \(y\) to get the equation of the tangent line: \(y - \frac{3}{2} = -\frac{3}{4}x + \frac{3}{4}\) \(y = -\frac{3}{4}x + \frac{3}{4} + \frac{3}{2}\) \(y = -\frac{3}{4}x + \frac{5}{2}\) Thus, the equation of the tangent line to the graph of \(f(x) = \frac{3}{2x}\) at the point \((1, \frac{3}{2})\) is \(y=-\frac{3}{4}x + \frac{5}{2}\).

Key Concepts

Tangent Line

In the realm of calculus, the tangent line is a fundamental concept that describes the straight line that touches a curve at a single point without crossing it. It essentially "hugs" the curve at that exact spot, providing an approximation of the curve's geometry at that point.
When you think about it, the tangent line tells us about the nature of the curve. It's like peeking into the curve's behavior around that particular point.
To find a tangent line to a curve at a point, we need two pieces of information:

• The exact point on the curve where the line will touch.
• The slope of the tangent line at that point.

The slope of the tangent line gives us the steepness and the direction that the line takes. With these, we can construct the equation of the tangent line using the point-slope formula:

• \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is the point.

Understanding tangent lines helps us in various practical applications, like calculating velocities and optimizing functions.

Slope of a Curve

The slope of a curve is crucial when working with derivatives and tangents. But what exactly is it?
Simply put, the slope at a specific point on a curve is the rate at which the curve rises or falls. It's the same idea as the slope of a straight line but applied to a nonlinear graph.

• For curves, this involves solving the derivative of the function, which tells us the slope at any given point.
• In cases where a curve changes its steepness or direction, the slope varies depending on where you are on the curve.

To find the slope of a curve at a point, follow these steps:

• Calculate the derivative of the function to get a formula representing the slope for all points.
• Substitute the given point's x-coordinate into the derivative to find the slope at that specific location.

For example, with \(f(x) = \frac{3}{2x}\), the derivative \(f'(x) = -\frac{3}{4x^2}\) represents the slope of the curve everywhere on the graph. To find the slope at \(x = 1\), plug \(1\) into the derivative, resulting in \(-\frac{3}{4}\). It reveals that the curve is decreasing at that point.

Power Rule

When it comes to differentiating functions, the power rule is one of the simplest and most efficient techniques in calculus.
If you have a function of the form \(x^n\), the power rule helps you quickly find its derivative.
The process involves the following steps:

• Multiply the exponent by the coefficient in front of the term.
• Decrement the exponent on the variable by one.

Using the power rule, the derivative of \(x^n\) is \(nx^{n-1}\).
Consider the function \(f(x) = \frac{3}{2x} = 3(2x)^{-1}\). Applying the power rule, recognize that the negative exponent can be handled just like any other exponent. Multiply and reduce the power:

• Multiply 3 by -1, resulting in -3.
• Decrease the power from -1 to -2, eliminating one more from the exponent.
• Simplify the expression to: \(-\frac{3}{4x^2}\).

Thus, the power rule simplifies finding derivatives, making it indispensable for understanding how functions behave.