Problem 9 - Active Calculus

Question

For each prompt that follows, sketch a possible graph of a function on the interval \(-3<\) \(x<3\) that satisfies the stated properties. a. \(y=f(x)\) such that \(f\) is increasing on \(-3

Step-by-Step Solution

Answer

Part a: Increasing, concave up (left of 0), concave down (right of 0); Part b: Increasing, concave down (left of 0), concave up (right of 0); Part c: Decreasing; concave up (left of -1), linear (-1 to 1); concave down (right of 1); Part d: Decreasing and concave down (left of 0), increasing and concave down (right of 0).

Step 1: Understanding the Function Properties

For each prompt, analyze the given properties such as increasing, decreasing, concave up, and concave down. This will guide the shape of the graph.

Step 2: Sketch for Part (a)

For function \(f(x)\): - It is increasing on the interval \(-3 < x < 3\).- Concave up on \(-3 < x < 0\).- Concave down on \(+0 < x < 3\).To sketch this:- Draw a curve that rises from left to right (increasing).- From \(x = -3\) to \(x = 0\), the curve should be concave up (shaped like an upward-facing bowl).- From \(x = 0\) to \(x = 3\), the curve should be concave down (shaped like a downward-facing bowl).

Step 3: Sketch for Part (b)

For function \(g(x)\): - It is increasing on the interval \(-3 < x < 3\).- Concave down on \(-3 < x < 0\).- Concave up on \(+0 < x < 3\).To sketch this:- Draw a curve that rises from left to right (increasing).- From \(x = -3\) to \(x = 0\), the curve should be concave down (shaped like a downward-facing bowl).- From \(x = 0\) to \(x = 3\), the curve should be concave up (shaped like an upward-facing bowl).

Step 4: Sketch for Part (c)

For function \(h(x)\): - It is decreasing on the interval \(-3 < x < 3\).- Concave up on \(-3 < x < -1\).- Neither concave up nor down on \-1 < x < 1\.- Concave down on \(+1 < x < 3\).To sketch this:- Draw a curve that falls from left to right (decreasing).- From \(x = -3\) to \(x = -1\), the curve should be concave up (shaped like an upward-facing bowl).- From \(x = -1\) to \(x = 1\), the curve should be linear (neither concave up or down).- From \(x = 1\) to \(x = 3\), the curve should be concave down (shaped like a downward-facing bowl).

Step 5: Sketch for Part (d)

For function \(p(x)\): - It is decreasing on the interval \-3 < x < 0\.- Concave down on \-3 < x < 0\.- Increasing on the interval \0 < x < 3\.- Concave down on \0 < x < 3\.To sketch this:- From \(x = -3\) to \(x = 0\), draw a curve that falls from left to right (decreasing) and is concave down (shaped like a downward-facing bowl).- From \(x = 0\) to \(x = 3\), draw a curve that rises from left to right (increasing) and is also concave down (shaped like a downward-facing bowl).

Key Concepts

increasing and decreasing functions

Understanding when a function is increasing or decreasing is fundamental in calculus. An increasing function means that as you move from left to right on the x-axis, the y-values of the function are going up. Conversely, a decreasing function means that the y-values are going down as you move from left to right.

Here's a simple rule to remember: if the derivative of the function, noted as \(f'(x)\), is positive over an interval, the function is increasing on that interval. If \(f'(x)\) is negative, the function is decreasing.

For example, in the exercises above:

• Function \(f(x)\) is increasing on \(-3 < x < 3\) because its derivative is positive throughout that interval.
• Function \(h(x)\) is decreasing on \(-3 < x < 3\) as its derivative is negative.
• Function \(p(x)\) decreases when \(-3 < x < 0\) and increases when \(0 < x < 3\), showcased by its derivative changing sign at \(x = 0\).

concavity

Concavity refers to the direction a graph is bending. Concavity can be concave up (shaped like a 'cup') or concave down (shaped like a 'cap').

To determine concavity, we look at the second derivative of the function, noted as \(f''(x)\). If \(f''(x) > 0\), the function is concave up. If \(f''(x) < 0\), the function is concave down.

In the given exercises:

• Function \(f(x)\) is concave up on \(-3 < x < 0\) because \(f''(x) > 0\) and concave down on \(0 < x < 3\) as \(f''(x) < 0\).
• Function \(g(x)\) follows the opposite pattern: concave down on \(-3 < x < 0\) and concave up on \(0 < x < 3\).
• Function \(h(x)\) is a bit more complex: it's concave up on \(-3 < x < -1\), neither care up nor down on \(-1 < x < 1\) (implying \(f''(x) = 0\)), and concave down on \(1 < x < 3\).
• Function \(p(x)\) is concave down throughout both intervals: \(-3 < x < 0\) and \(0 < x < 3\), showcasing consistent negative second derivatives.

graph sketching

To sketch a graph, it's critical to understand the properties of the function, like where it increases, decreases, and its concavity.

Here are some tips for sketching graphs based on the given exercise:

• Start by marking key intervals and points, such as the points where the function changes concavity or behavior.
• Use the information about increasing or decreasing behavior to guide the general direction of your sketch (rising or falling).
• Apply concavity to shape the curve appropriately (concave up or down).

For example, in part (a), \(y=f(x)\):
- Draw a curve that increases from left to right because the function is increasing over the whole interval.
- From x = -3 to x = 0, make the curve concave up (shaped like an upward-facing bowl).
- From x = 0 to x = 3, make the curve concave down (shaped like a downward-facing bowl).

These steps will help ensure your graph accurately represents the function's behavior across the interval.