Problem 49 - Calculus an Applied Approach

Question

(a)Find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ {y=-2 x^{4}+5 x^{2}-3} \quad (1,0) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the graph of the function at the point (1,0) is \(y = 2x - 2\). Use a graphing utility to graph the function and its tangent line and the derivative feature of a graphing utility to confirm these results.

Step 1: Computing the derivative of the function

Let's start by finding the derivative of the function. For a given function \(y=f(x)\), the derivative \(f'(x)\) is computed as follows: \(f'(x) = -8x^{3}+10x\). We obtained this by applying the power rule of derivatives, which states that the derivative of \(x^n\), where \(n\) is any real number, is \(n*x^{n-1}\).

Step 2: Finding the slope of the tangent at the given point

The derivative of a function at a particular point gives the slope of the tangent at that point. So, we substitute \(x=1\) into the derivative function to find the slope of the tangent \(m\) at the given point (1,0). Thus, \(f'(1) = -8*1^3 + 10*1 = -8+10 = 2\).

Step 3: Finding the equation of the tangent

We now have a point (1, 0) and a slope \(m=2\), which are sufficient to find the equation of the tangent line. The formula for a straight line is \(y - y_{1} = m(x - x_{1})\), where \((x_{1}, y_{1})\) is a point on the line and \(m\) is the slope of the line. Substituting the values, we get the equation of the tangent line as: \(y - 0 = 2(x - 1)\), which simplifies to \(y = 2x - 2\).

Key Concepts

Understanding Derivatives

Derivatives are a key concept in calculus, essential for understanding how functions change. In simple terms, the derivative of a function at a point gives the rate at which the function's value is changing at that point. For example, if you have a graph of a function, the derivative at a specific point tells you the slope of the tangent line at that point. This is crucial for predicting behavior and trends in various fields like physics, economics, and engineering.
To compute a derivative, you often rely on various rules and techniques. One of the most common ones is the power rule, which simplifies the differentiation process. By understanding derivatives and their calculations, you can solve many mathematical and real-world problems efficiently.

Applying the Power Rule

The power rule is a very handy shortcut for finding derivatives of polynomials. It states that if you have a term of the form \(x^n\), its derivative will be \(n*x^{n-1}\). This means you simply multiply the coefficient by the exponent and then decrease the exponent by one.
For example, in the function \(y = -2x^4 + 5x^2 - 3\), applying the power rule will give us \(-8x^3\) from \(-2x^4\) and \(10x\) from \(5x^2\). The derivative, \(f'(x) = -8x^3 + 10x\), is then used to find the slope of the tangent line at any given point. By mastering the power rule, you will significantly speed up the process of finding derivatives, especially when dealing with polynomials.

Slope of a Tangent Line

The slope of a tangent line is vital for understanding how a curve behaves at a particular moment. When you compute the derivative of a function at a specific point, you’re essentially finding this slope.
In the context of the given problem, by substituting \(x = 1\) into the derivative \(f'(x) = -8x^3 + 10x\), we calculated the slope at the point (1,0) as 2. This result indicates that at \(x = 1\), the function is increasing at the rate of two units vertically for every unit it moves horizontally.
The tangent line, therefore, gives a perfectly "linear approximation" of the function at that neighborhood. Knowing the slope allows you to write the equation of the tangent line, which in this exercise simplifies to \(y = 2x - 2\).

Using a Graphing Utility

Graphing utilities are powerful tools, helping to visualize functions and their derivatives easily. By plotting a function and its tangent line, you can verify the calculations made by hand. Use the utility's derivative feature to confirm the slope of the tangent line at a specific point.
For this task, by inputting the original function \(y = -2x^4 + 5x^2 - 3\) and its tangent line equation \(y = 2x - 2\) into the graphing utility, you can see how the tangent line just touches the curve at the point (1,0).
This visualization helps reinforce your understanding of how derivatives and tangent lines work in practice. Next time you're solving for tangent lines or derivatives, remember that graphing utilities can provide both confirmation and a deeper intuition of the problem at hand.