Problem 19 - 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. \(f(x)=3 x^{2}\) at \((1,3)\)

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the function \(f(x)=3x^2\) at the point \((1,3)\) is 6. The equation of the tangent line at this point is \(y=6x-3\).

Step 1: Find the derivative of the function

To find the slope of the tangent line, we need to find the derivative of the function. The derivative of a function is the rate of change of the function, which gives us the slope of the function at any given point. For the function \( f(x) = 3x^2 ,\) the derivative is given by: \( f'(x) = \frac{d}{dx}(3x^2) = 6x \) 2)

Step 2: Evaluate the derivative at the given point

Now we need to evaluate the derivative at the given point. Since the point is \((1, 3),\) we want to find the slope of the tangent line at \(x = 1.\) So, we just need to plug in \(x = 1\) into our derivative: \(f'(1) = 6(1) = 6\) This means the slope of the tangent line at the point \((1, 3)\) is 6. 3)

Step 3: Determine the equation of the tangent line using point-slope form

Now that we have the slope of the tangent line and the given point \((1, 3),\) we can determine the equation of the tangent line using the point-slope form of a linear equation: \(y - y_1 = m(x - x_1)\) Plug in the given point \((x_1,y_1) = (1, 3)\) and the slope \(m = 6\): \(y - 3 = 6(x - 1)\) 4)

Step 4: Simplify the equation

Finally, we can simplify the equation of the tangent line: \(y - 3 = 6x - 6\) \(y = 6x - 3\) So the equation of the tangent line to the graph of the function \(f(x) = 3x^2\) at the point \((1, 3)\) is \(\boxed{y = 6x - 3}\).

Key Concepts

Tangent Line

The tangent line to a curve at a given point is a straight line that just "touches" the curve at that point, without crossing it. Imagine a smooth hill with a single bicycle tire gently leaning against the side. The point where the tire touches the hill is similar to how a tangent line touches a curve: at precisely one point. This line represents the immediate direction in which the curve is heading at that point. A tangible property of the tangent line is its slope, which numerically tells us how steeply the line inclines or declines. This steepness changes from point to point on a curve, as curves are not straight lines. Therefore, pinpointing the slope of the tangent line is crucial when working with curves, offering a snapshot of the curve’s behavior at a specific point. For the function given in the exercise, the tangent line's slope at point \((1, 3)\) is found using calculus, specifically, the derivative.

Derivative

The derivative of a function, in smooth layman’s terms, is a way to evaluate how a function changes at any given point. It serves the role of a "slope detective." For a straight line, the slope stays constant. But for curves, the slope is ever-changing. That's where derivatives come to the rescue—they reveal the slope at any particular point on the curve. To find the derivative of \( f(x) = 3x^2 \), you differentiate the function with respect to \( x \). In this case, the derivative is \( f'(x) = 6x \). This tells us that the slope of the tangent line varies according to \( 6x \). In practice, you plug the \( x \)-value of the point of interest, \((1, 3)\) in the exercise, into this derivative to find that the slope at \( x = 1 \) is \( 6 \). Therefore, derivatives are not just mathematical instruments; they are tools for interpreting the varying dynamics of a curve.

Point-Slope Form

Once the slope of the tangent line is known, crafting the equation for this line becomes straightforward. The point-slope form of a line is a nifty formula used to write the equation of a line when you know the line’s slope and one point on the line. This form is captured in the equation: - \( y - y_1 = m(x - x_1) \) Where: - \( m \) represents the slope, - \( (x_1, y_1) \) represents a point on the line.In the exercise, the point \((1, 3)\) and the slope \( 6 \) lead to the equation: - \( y - 3 = 6(x - 1) \)You then simplify this to find the standard linear equation for the line: - \( y = 6x - 3 \)Understanding the point-slope form is super helpful; it gracefully marries the slope and a point to birth the equation of a line.