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 \(g\) '. $$ \begin{aligned} &f(x)=x^{2} \\ &g(x)=3 x^{2} \end{aligned} $$

Short answer

The derivatives of the functions at \(x=1\) are \(f'(1) = 2\) and \(g'(1) = 6\). The function \(g\) is essentially the function \(f\) multiplied by 3, so the derivative of \(g\) is always 3 times the derivative of \(f\) at any value of \(x\).

Step-by-step solution

Graph the Functions

First, plot the provided functions \(f(x)=x^{2}\) and \(g(x)=3x^{2}\). Both are quadratic functions, with \(f\) being a basic quadratic function which opens upwards and \(g\), being an stretched version of the basic quadratic function by a factor of 3.

Calculate the Derivatives

Next, find the derivatives of the functions \(f(x)\) and \(g(x)\) to get \(f'(x) = 2x\) and \(g'(x) = 6x\). Then, substitute \(x=1\) into these functions to calculate \(f'(1) = 2\) and \(g'(1) = 6\). These are the slopes of the tangent lines to the graphs of \(f\) and \(g\) at \(x=1\).

Sketch the Tangent Lines

Then sketch the tangent lines to the graphs of \(f\) and \(g\) at \(x=1\). These lines will have slopes of 2 and 6 respectively. The tangent line can be visualized as a straight line that touches the graph of a function at one point without crossing through it.

Explain the Relationship

Finally, explain the relationship between \(f'\) and \(g\). The functions \(g'(x) = 6x\) gives the rate of change of \(g\) at each \(x\) and it coincides to be 3 times \(f'(x) = 2x\), the rate of change of \(f\). Hence, observe due to \(g(x) = 3f(x)\), the derivative of \(g\) will always be 3 times the derivative of \(f\) at the same \(x\).

Key concepts

Understanding Quadratic Functions

Quadratic functions are a foundational element in calculus and algebra, characterized by an equation of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the example exercise with the function \( f(x) = x^2 \) and \( g(x) = 3x^2 \), both equations represent quadratic functions.

The graph of any quadratic function is a parabola, a symmetrical curve that opens upwards if \( a > 0 \) or downwards if \( a < 0 \). For instance, \( f(x) = x^2 \) generates a standard parabola opening upwards with the vertex at the origin (0,0). When we introduce a multiplicative factor, as seen in \( g(x) = 3x^2 \), the parabola remains symmetrical but becomes narrower due to the stretching effect of the coefficient 3.

To better understand quadratic functions, one can visualize them by plotting points and observing their U-shaped curves. The importance of mastering quadratic functions lies in their predictability and the plethora of real-world applications they model, such as projectile motion and the optimization of area and profit in businesses.

Grasping Derivatives

When diving into calculus, derivatives represent an essential tool, providing the rate at which a function is changing at any given point. They are the cornerstone of calculus education and can be thought of as the 'instantaneous rate of change' or the slope of the tangent line at a particular point on a graph.

In our textbook problem, the functions provided are \( f(x) = x^2 \) and \( g(x) = 3x^2 \). Their derivatives, \( f'(x) = 2x \) and \( g'(x) = 6x \), respectively, indicate how steep the functions are at any point \( x \). The concept of a derivative is not just a formula; it encapsulates movement and change, key ideas that permeate through all fields of science, economics, and beyond.

Understanding how to compute and interpret derivatives is a monumental step in appreciating the dynamic nature of the world, where almost everything is in continuous motion or transformation. Through the derivative, we quantify change, enabling us to predict and adapt to various situations.

The Role of Tangent Lines

Tangent lines are a visual representation of the abstract concept of derivatives. They are straight lines that touch a curve at precisely one point without intersecting it at that instant. The slope of a tangent line to a curve at a given point reflects the instantaneous rate of change of the function at that point; in other words, the derivative.

In the exercise given, sketching tangent lines to the graphs of \( f(x) = x^2 \) and \( g(x) = 3x^2 \) at \( x=1 \) means drawing lines that just 'kiss' the curves at the points where \( x=1 \). The slopes of these lines, as we've calculated, are 2 and 6, respectively, which match the derivatives at that point.

Tangent lines are not just a theoretical concept, they serve as a practical tool for approximating function values near the point of tangency, greatly simplifying complex analyses. Whether in geometry or in studying real-world systems, understanding tangent lines enriches one's ability to analyze and predict behavior at any given moment.