Problem 58 - Calculus an Applied Approach

Question

(a) sketch the graphs of \(f\) and \(g,(b)\) find \(f^{\prime}(1)\) and \(g^{\prime}(1),(c)\) sketch the tangent line to each graph when \(x=1,\) and \((d)\) explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). $$ \begin{array}{l}{f(x)=x^{2}} \\ {g(x)=3 x^{2}}\end{array} $$

Step-by-Step Solution

Answer

The graphs of \(f(x)=x^2\) and \(g(x)=3x^2\) are parabolas, with \(g\) being a steeper graph. The derivatives of both functions at \(x=1\) are \(f'(1) = 2\) and \(g'(1) = 6\). Therefore, \(g'(x) = 3f'(x)\), which means the slope of \(g\) at any point is three times the slope of \(f\) at the same point.

Step 1: Sketch the graphs of f and g

To sketch graph of \(f(x)=x^2\), plot a few points and join them. For instance, for \(x=-2, -1, 0, 1, 2\), \(y\) values are \(4, 1, 0, 1, 4\) respectively. Similarly, for \(g(x)=3x^2\), plot points for same \(x\) values which yields \(y\) as \(12, 3, 0, 3, 12\). The graph of \(g\) would be similar to \(f\) but steeper due to multiplication by 3.

Step 2: Find the derivatives of f and g at point x=1

The derivative of a function at a certain point shows the slope of the tangent to the graph of the function at that point. Using power rule, derivative \(f'(x) = 2x\) and \(g'(x) = 6x\). Plugging \(x=1\) we find that \(f'(1) = 2\) and \(g'(1) = 6\). This means that at \(x=1\), slope of tangent line to \(f\) is 2 and to \(g\) is 6.

Step 3: Sketch the tangent line to each graph at x=1

With \(f'(1) = 2\) and \(g'(1) = 6\), the lines tangent to the functions \(f(x)\) and \(g(x)\) at \(x=1\) have slopes of 2 and 6 respectively. The equation of tangent line is given by \(L(x) = f'(a)(x-a) + f(a)\) where \(f'(a)\) is the slope and \(a\) is the point of tangency. Here, for \(f(x)\), \(L(x) = 2(x-1) + 1 = 2x-1\), and for \(g(x)\), \(L(x) = 6(x-1) + 3 = 6x-3\). So, the steepness of the tangent to \(g(x)\) at \(x=1\) is greater than the steepness of the tangent to \(f(x)\) at \(x=1\).

Step 4: Relate the derivatives \(f'(x)\) and \(g'(x)\)

Comparing \(f'(x) = 2x\) and \(g'(x) = 6x\), thus, \(g'(x)=3f'(x)\). Essentially, this means the slope of the tangent line of \(g\) at any point \(x\) will be three times the slope of the tangent line of \(f\) at that same point.

Key Concepts

Tangent Lines

Tangent lines are a fundamental concept in calculus, representing straight lines that touch a curve at just one specific point. This point is where the slope of the tangent matches the slope of the curve. The tangent line to a curve at a given point can be thought of as the best linear approximation to the curve at that point.

To calculate the equation of a tangent line to a function at a particular point, you need two pieces of information:

• The point where the tangent line touches the curve.
• The slope of the tangent line at that point, which is the derivative of the function at this point.

For example, given a function like \(f(x)=x^2\), finding the tangent line at \(x=1\) involves calculating the derivative \(f'(x)\), which tells us the slope of the tangent line, and using it in the equation form \(L(x) = f'(a)(x-a) + f(a)\) to find the actual line equation.

Calculus

Calculus is a branch of mathematics that focuses on rates of change and the accumulation of quantities. At its core are two main operations: differentiation and integration. In this problem, we focus on differentiation, which involves finding the derivative of a function.

The derivative of a function at a given point provides valuable information about the behaviour of the function at that point. This includes the rate at which the function's value changes as its input changes, which we often refer to as the function's slope.

• Differentiation allows us to determine the slope of tangent lines, which is crucial for understanding how a function behaves locally.
• By using derivatives, we can predict how functions increase or decrease, helping us study systems in real-world applications, such as physics and engineering.

The derivative is a key concept not only in understanding tangent lines but also in determining how functions react over different intervals.

Power Rule

The power rule is a simple and widely-used technique in calculus for finding the derivative of functions that have terms of the form \(x^n\). The rule states that the derivative of \(x^n\) is \(nx^{n-1}\). This is particularly useful for polynomial functions, which are common in various mathematical problems.

Applying the power rule makes differentiation straightforward:

• For \(f(x) = x^2\), using the power rule gives \(f'(x) = 2x\).
• For \(g(x) = 3x^2\), it leads to \(g'(x) = 6x\).

This rule effectively makes it easy to compute how functions change and to find the slopes of tangent lines quickly. In the exercise, the power rule helped determine the slopes of the tangent lines at \(x=1\), where we found \(f'(1) = 2\) for \(f(x)\), and \(g'(1) = 6\) for \(g(x)\). Understanding this rule is fundamental for students working on calculus problems.

Function Graphs

Visualizing functions through their graphs is an integral part of understanding their behavior. Graphs offer a visual representation of the function's output in relation to its input, providing insights into the function's characteristics, such as where it increases or decreases, and its curvature.

For example, in the function \(f(x) = x^2\), the graph is a parabola opening upwards. This visually demonstrates how the function behaves:

• The curve is symmetric about the y-axis, showing it's an even function.
• The graph of \(g(x) = 3x^2\) shares the same shape but is steeper, tripling the y-values of \(f(x)\) for any given x, because the coefficient in front of \(x^2\) is 3.

Graphs also allow us to plot tangent lines, providing instant information about the slope and direction of the function at specific points. By sketching the functions, students can better grasp the relationship between the algebraic expressions and their graphical representations.