Problem 1 - Calculus A New Horizon

Question

In each part, sketch the graph of a function \(f\) with the stated properties, and discuss the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) (a) The function \(f\) is concave up and increasing on the interval \((-\infty,+\infty)\) (b) The function \(f\) is concave down and increasing on the interval \((-\infty,+\infty)\) (c) The function \(f\) is concave up and decreasing on the interval \((-\infty,+\infty)\) (d) The function \(f\) is concave down and decreasing on the interval \((-\infty,+\infty)\)

Step-by-Step Solution

Answer

(a) \(+,+\), (b) \(-,+\), (c) \(+,-\), (d) \(-,-\).

Step 1: Understanding Concavity and Monotonicity

A function is concave up if its second derivative, \( f''(x) \), is positive, and concave down if \( f''(x) \) is negative. Similarly, a function is increasing if its first derivative, \( f'(x) \), is positive, and decreasing if \( f'(x) \) is negative.

Step 2: Sketching Graph for Part (a)

For part (a), the graph should be a curve that bends upwards and goes from the bottom left to the top right of the graph. Since the function is concave up, \( f''(x) > 0 \), and since it is increasing, \( f'(x) > 0 \). An example could be \( f(x) = e^x \).

Step 3: Sketching Graph for Part (b)

For part (b), the graph should be a curve that bends downwards but still moves upwards overall across the graph. Since it is concave down, \( f''(x) < 0 \), and since it is increasing, \( f'(x) > 0 \). An example could be \( f(x) = \ln(x) \), with appropriate domain adjustments for positivity.

Step 4: Sketching Graph for Part (c)

For part (c), the graph should be a curve that bends upwards but moves downwards overall across the graph. The function is concave up, so \( f''(x) > 0 \), but since it is decreasing, \( f'(x) < 0 \). An example of this is \( f(x) = -e^x \).

Step 5: Sketching Graph for Part (d)

For part (d), the graph should be a curve bending downwards and moving downwards across the graph. Since it is concave down, \( f''(x) < 0 \), and since it is decreasing, \( f'(x) < 0 \). An example could be \( f(x) = -\ln(x) \), assuming appropriate constraints on its domain.

Step 6: Summarizing the Signs for Each Part

(a) Concave up and increasing: \( f''(x) > 0 \), \( f'(x) > 0 \). (b) Concave down and increasing: \( f''(x) < 0 \), \( f'(x) > 0 \). (c) Concave up and decreasing: \( f''(x) > 0 \), \( f'(x) < 0 \). (d) Concave down and decreasing: \( f''(x) < 0 \), \( f'(x) < 0 \).

Key Concepts

Concavity

Concavity describes how the curve of a function behaves as it moves along its graph. If you imagine a bowl, the direction in which it opens can help you understand concavity.A function is said to be "concave up" if it opens upwards like a smile. In this case, the second derivative of the function, represented as \( f''(x) \), is greater than zero \( f''(x) > 0 \). A concave up graph might look like the curve of an exponential function growing upwards.On the flip side, a function is "concave down" if it opens downwards like a frown. Here, the second derivative \( f''(x) \) is less than zero \( f''(x) < 0 \). This might be seen in a downward-opening parabola.

Monotonicity

Monotonicity refers to the behavior of a function in terms of increasing or decreasing. A function is increasing when its graph goes upwards as you move from left to right. The first derivative \( f'(x) \) for an increasing function is positive \( f'(x) > 0 \).Conversely, a function is decreasing when its graph moves downwards as you travel from left to right. In this scenario, \( f'(x) \) is negative \( f'(x) < 0 \).Understanding monotonicity is crucial when sketching graphs because it helps determine the general direction a function moves.

Derivatives

Derivatives are a fundamental tool in calculus that provide significant information about a function's behavior.

• The first derivative, \( f'(x) \), indicates the slope of the tangent line to the graph at any given point. It tells us whether a function is increasing or decreasing.
• The second derivative, \( f''(x) \), provides insights into the concavity of the function. It explains whether the curve is bending upwards or downwards.

Calculating these derivatives allows us to understand and predict the pattern of the function without graphing it fully. This is especially useful when you need to make quick assessments about the shape and behavior of the function.

Function Graph Properties

The properties of a function graph encompass both its shape and movement along the axes.

• If a graph is concave up and increasing continuously, it will have a curve that goes upwards like an exponential growth.
• If it's concave down but increasing, the graph will seem to climb upward but with a downward bend, showing diminishing returns as it rises.
• A concave up and decreasing graph will dip, creating a downward curve that still maintains an upward bent like an inverted exponential function.
• Finally, a concave down and decreasing graph will slope down with an added downward curve, resembling a bowl turned on its side moving downward.

These graph properties are visual cues that help you quickly recognize and interpret the behavior of a function's graph effortlessly.