Question

Group Activity In Exercises \(39-42,\) sketch a graph of a differentiable function \(y=f(x)\) that has the given properties. $$ \begin{array}{l}{\text { (a) local minimum at }(1,1), \text { local maximum at }(3,3)} \\ {\text { (b) local minima at }(1,1) \text { and }(3,3)} \\\ {\text { (c) local maxima at }(1,1) \text { and }(3,3)}\end{array} $$

Short answer

Sketch three separate graphs, each with the points (1,1) and (3,3): graph A should dip down to (1,1) and peak at (3,3), graph B should have two dips at these respective points, and graph C should have two peaks at these points.

Step-by-step solution

Sketching Graph A

For graph A, begin by marking the points (1,1) and (3,3) on your graph. Now, sketch a curve that dips down to (1,1), forming a 'U' shape, which represents a local minimum. Then, have the curve rise up and peak at (3,3), forming an 'n' shape, which represents a local maximum. The curve should be smooth and continuous, as the function is differentiable.

Sketching Graph B

For graph B, begin by marking the points (1,1) and (3,3) on your graph. Sketch two separate 'U' shapes, one at each of these points, to indicate the local minima. Remember to connect these two shapes with a smooth curve, as the function is differentiable.

Sketching Graph C

For graph C, begin by marking the points (1,1) and (3,3) on your graph. Sketch two separate 'n' shapes, one at each of these points, to indicate the local maxima. As with the previous graphs, connect these two shapes with a smooth curve to maintain the function's differentiability.

Key concepts

Differentiable Function

When discussing differentiable functions in calculus, we're dealing with a foundational concept that bridges algebra and calculus. A function is said to be differentiable at a point if it has a derivative there. This means that the function has a defined slope at that point — the tangent line to the function's curve is not vertical, and the curve has no sharp corners or cusps.

Mathematically, if a function is differentiable at a point, it implies that it's also continuous there, but the reverse isn't always true. When graphing, a differentiable function will appear as a smooth curve with no breaks or sharp angles. The importance of a function being differentiable cannot be overstated, as this property allows us to utilize calculus tools like differentiation and integration to analyze and predict the behavior of functions with accuracy.

For instance, when creating a graph of a differentiable function that presents a local minimum at \(1,1\) and a local maximum at \(3,3\), we ensure the graph dips to form a 'U' at \(1,1\) and peaks with an 'n' shape at \(3,3\), all the while maintaining a smooth curve that doesn't have breaks or sharp turns, to visualize the function's differentiability.

Local Minimum and Maximum

The terms local minimum and local maximum are crucial when analyzing the behavior of functions. These terms refer to points on the graph of a function where the function values are lower or higher than those immediately surrounding them, respectively. More intuitively, at a local minimum, the graph forms a valley, a low point surrounded by higher points. Conversely, at a local maximum, the graph creates a peak, being higher than the neighboring areas.

To sketch these concepts, you might think of the classic 'U' shape for minimums and an 'n' shape for maximums within their localized regions. This concept is particularly crucial because local extrema (minima and maxima) often represent important real-world values when applying calculus to physics, economics, engineering, etc. For example, they can signify the least cost, greatest efficiency, highest pressure, and so on.

Improvement Activity

To explain this in a classroom or tutorial setting, consider presenting real-world scenarios that require finding local minima and maxima, such as optimizing business profits or determining the best position for a satellite in orbit.

Graphical Representation of Calculus Concepts

The graphical representation of calculus concepts is a powerful tool for visual learners. By drawing the graphs of functions, students can see the abstract concepts of calculus — such as limits, continuity, derivatives, and integrals — come to life. These visual cues are not only aids for understanding but also for retaining the information.

When teaching calculus, exhibiting these graphs alongside the algebraic formulas breathes life into the numbers and variables. For instance, when a function's graph shows a tangent line at any given point, students can witness the derivative in action. Alternatively, the area under a curve demonstrates integration visually.

To sketch these ideas accurately, remember to discuss why certain features appear on a graph. Explain how the smooth nature of curves represents differentiable functions, or how the 'U' and 'n' shapes are indicative of local minima and maxima. Involve the students by having them analyze how these shapes evolve as functions change and ensuring their understanding of what these graphs represent in a wider mathematical and real-world context.