Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=1 ; f(6)=3\); increasing and concave down on \((0.6)\)

Step-by-Step Solution

Answer

Draw a continuous graph from \((0,1)\) to \((6,3)\) that is increasing and concave down.

Step 1: Analyze Conditions

We need to draw a continuous function defined on the interval \([0,6]\) with the specific properties: \(f(0) = 1\), \(f(6) = 3\), increasing throughout \((0,6)\), and concave down in the interval \((0,6)\).

Step 2: Plot Initial and Final Points

Begin by plotting the points \((0,1)\) and \((6,3)\) on a graph. These points satisfy the condition \(f(0)=1\) and \(f(6)=3\).

Step 3: Ensure Increasing Nature

Since the function must be increasing on \((0,6)\), draw the rest of the graph such that it steadily rises from \((0,1)\) to \((6,3)\). No part of the graph should decrease.

Step 4: Implement Concave Down Behavior

For the function to be concave down on \((0,6)\), the slope must continuously decrease as you move from left to right across the interval. Visually, the graph should look like a downward-facing arch or part of a parabola opening downwards.

Step 5: Verify Conditions

After drawing the graph, double-check that at no point in \((0,6)\) does the graph decrease and that it maintains the concavity. Ensure the graph passes through the points \((0,1)\) and \((6,3)\).

Key Concepts

Concave Functions

Concave functions are essential for understanding how a curve bends. If a function is concave down, its graph resembles an upside-down bowl. This means any line segment connecting two points on the graph will lie below the curve itself.

In mathematical terms, for a function to be concave down on an interval, the second derivative must be less than or equal to zero. This ensures that the rate of increase of the function's slope decreases as you move along the graph.

• Visual appearance: Downward sloping curve.
• Second derivative: Less than or equal to zero.
• Line segment property: Lies below the curve.

Understanding concave functions helps you anticipate the shape and direction of a graph, simplifying the process of graph sketching.

Graph Sketching

Graph sketching combines art and math to visually represent functions based on certain properties. Here, the goal is to sketch a function over an interval while fulfilling the conditions of continuity, specific points, and behaviors like increasing or concave down.

Start by plotting essential points given by the function, such as where it intersects the axes or known specific points, like \( (0,1) \) and \( (6,3) \) in our example. After marking these crucial points, pay close attention to the function's characteristics that dictate its shape between these points.

• Identify and plot critical points.
• Consider continuity: No breaks or jumps are allowed.
• Ensure behavior fits the given characteristics (e.g., increasing, concave down).

Looking at these factors ensures the sketch reflects the true mathematical properties required for accurate analysis of the function.

Increasing Functions

An increasing function rises as you move from left to right along the x-axis. This means the y-values of the function grow larger as the x-values increase. Ensuring this behavior can be visualized by ensuring the slope of the line segment remains positive across the domain.

It's crucial when sketching an increasing function not to let any part of the graph dip downwards. For example, in our given interval, the entire function must gently ascend from \( (0, 1) \) to \( (6, 3) \), never decreasing at any point.

• Slope of the tangent: Positive along the entire interval.
• Visual goal: Consistently ascending curve.
• Behavior check: No dips or downward sections.

This increasing pattern ensures that you meet the specified conditions and correctly represent positive growth over the interval.