Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch a graph of a continuous function defined for all real numbers with all three characteristics listed. If it is impossible to do this, say so, and draw the graph of a function with the three characteristics on the domain \([-1,1]\). (a) \(f\) is positive, increasing, and concave up (b) \(f\) is positive, increasing, and concave down (c) \(f\) is negative, increasing, and concave up (d) \(f\) is negative, increasing, and concave down (e) \(f\) is positive, decreasing, and concave up (f) \(f\) is positive, decreasing, and concave down (g) \(f\) is negative, decreasing, and concave up (h) \(f\) is negative, decreasing, and concave down

Step-by-Step Solution

Answer

The graphs for the functions can be sketched using the listed properties. In all scenarios, the function's graph retains the given characteristics within the interval [-1,1].

Step 1: Sketch for function f(a)

The function is positive, increasing, and concave up. A plausible example could be the function \(f(x) = x^2\). It is positive for all real numbers, increasing as x increases and is concave up as the second derivative is positive.

Step 2: Sketch for function f(b)

The function is positive, increasing, and concave down. A possible function that suits these characteristics could be \(f(x) = -x^2 + 2x + 1\). This function is increasing in the interval \(-1,1\) and is positive as per given conditions, while its second derivative is negative indicating a concave down graph.

Step 3: Sketch for function f(c)

The function is negative, increasing, and concave up. An example could be the function \(f(x) = -x^2 + x - 1\). This function is negative, increasing over the interval [-1,1], and is concave up due to its positive second derivative.

Step 4: Sketch for function f(d)

The function is negative, increasing, and concave down. A plausible function could be \(f(x) = -x^2 + 2x - 2\). The function is negative, its value increases over the interval [-1,1], and its graph is concave down as the second derivative is negative.

Step 5: Sketch for function f(e)

The function is positive, decreasing, and concave up. A plausible function with these properties could be \(f(x) = -x^2 + 2x + 1\). It is positive, decreasing over the interval [-1,1], and is concave up as its second derivative is positive.

Step 6: Sketch for function f(f)

The function is positive, decreasing, and concave down. For example, the function \(f(x) = x^2 - 2x + 1\), is positive, decreases over the interval [-1,1], and its graph is concave down as the second derivative is negative.

Step 7: Sketch for function f(g)

The function is negative, decreasing, and concave up. A function that fits the description is \(f(x) = -x^2 - x - 1\). It is negative, decreases over the interval [-1,1], and its graph is concave up due to its positive second derivative.

Step 8: Sketch for function f(h)

The function is negative, decreasing, and concave down. For example, \(f(x) = x^2 + 2x - 1\). The function is negative, decreases over the interval [-1,1], and its graph concave down as the second derivative is negative.

Key Concepts

Understanding Concavity

Concavity is all about the shape of the graph. Picture the graph as a bowl. If the bowl opens upwards, like a smiling face, the graph is concave up. Mathematically, this is when the second derivative of a function, denoted as \(f''(x)\), is positive. On the other hand, if the bowl opens downwards, like a frown, the graph is concave down. Here, \(f''(x)\) is negative.
Concavity helps in understanding how the rate of change of a function itself is changing. It's like the slope of the slope! A graph that's concave up indicates that the rate of increase of the function is speeding up. Conversely, if it's concave down, any increase is slowing down. These concepts are particularly useful in understanding motion and optimization problems.

• Concave Up: \(f''(x) > 0\)
• Concave Down: \(f''(x) < 0\)

By analyzing the concavity, one can predict the turning points and how the function behaves around these points.

Increasing and Decreasing Functions

A function is increasing in a region if as \(x\) increases, the function values \(f(x)\) also rise. This typically means the slope of the tangent line is positive. In mathematical terms, the first derivative \(f'(x) > 0\), indicating an increasing function. If the function slopes downward, as \(x\) increases, then \(f(x)\) values decrease, and \(f'(x) < 0\), which is typical of a decreasing function.
Knowing whether a function is increasing or decreasing can tell us a lot about the graph. We can understand trends, predict long-term behavior in data and optimize certain quantities in practical applications.

• Increasing Function: \(f'(x) > 0\)
• Decreasing Function: \(f'(x) < 0\)

This concept helps in identifying intervals where functions achieve maximum or minimum values, crucial for solving many calculus problems.

Positive and Negative Functions

The concepts of positive and negative functions relate to the values of \(f(x)\) itself. A function is positive in an interval if \(f(x) > 0\) for all \(x\) in that interval. This means the graph lies above the x-axis in that range. Conversely, a negative function indicates \(f(x) < 0\), positioning the graph below the x-axis.
Understanding these regions of positivity and negativity can help in integrals to find areas, as well as understanding where a function might cross or not touch the axis again.

• Positive Function: \(f(x) > 0\)
• Negative Function: \(f(x) < 0\)

Knowing whether a function is positive or negative can provide insights into applied problems. For example, in economics, positive functions might denote profit, while negative functions could indicate loss.