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Chapter 1: Problem 9

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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).
03

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).
04

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).
05

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For each of the following prompts, give an example of a function that satisfies the stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such example is possible, explain why. a. A function \(f\) that is continuous at \(a=2\) but not differentiable at \(a=2\). b. A function \(g\) that is differentiable at \(a=3\) but does not have a limit at \(a=3\). c. A function \(h\) that has a limit at \(a=-2,\) is defined at \(a=-2,\) but is not continuous at \(a=-2\) d. A function \(p\) that satisfies all of the following: \- \(p(-1)=3\) and \(\lim _{x \rightarrow-1} p(x)=2\) $$\text { - } p(0)=1 \text { and } p^{\prime}(0)=0$$ \- \(\lim _{x \rightarrow 1} p(x)=p(1)\) and \(p^{\prime}(1)\) does not exist

Use linear approximation to approximate \(\sqrt{25.3}\) as follows. Let \(f(x)=\sqrt{x}\). The equation of the tangent line to \(f(x)\) at \(x=25\) can be written in the form \(y=m x+b\). Compute \(m\) and \(b\). \(m=\) \(\square\) \(b=\) \(\square\) Using this find the approximation for \(\sqrt{25.3}\). Answer: \(\square\)

Consider a car whose position, \(s\), is given by the table $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{~s}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ \hline s(\mathrm{ft}) & 0 & 0.45 & 1.7 & 3.8 & 6.5 & 9.6 \\ \hline \end{array}$$ Find the average velocity over the interval \(0 \leq t \leq 0.2\). average velocity \(=\square\) help (units) Estimate the velocity at \(t=0.2\) velocity \(=\square\) help (units)

A diver leaps from a 3 meter springboard. His feet leave the board at time \(t=0\), he reaches his maximum height of \(4.5 \mathrm{~m}\) at \(t=1.1\) seconds, and enters the water at \(t=2.45\). Once in the water, the diver coasts to the bottom of the pool (depth \(3.5 \mathrm{~m}\) ), touches bottom at \(t=7\), rests for one second, and then pushes off the bottom. From there he coasts to the surface, and takes his first breath at \(t=13\). a. Let \(s(t)\) denote the function that gives the height of the diver's feet (in meters) above the water at time \(t\). (Note that the "height" of the bottom of the pool is -3.5 meters.) Sketch a carefully labeled graph of \(s(t)\) on the provided axes in Figure 1.1.5. Include scale and units on the vertical axis. Be as detailed as possible. b. Based on your graph in (a), what is the average velocity of the diver between \(t=2.45\) and \(t=7 ?\) Is his average velocity the same on every time interval within [2.45,7]\(?\) c. Let the function \(v(t)\) represent the instantaneous vertical velocity of the diver at time \(t\) (i.e. the speed at which the height function \(s(t)\) is changing; note that velocity in the upward direction is positive, while the velocity of a falling object is negative). Based on your understanding of the diver's behavior, as well as your graph of the position function, sketch a carefully labeled graph of \(v(t)\) on the axes provided in Figure \(1.1 .6 .\) Include scale and units on the vertical axis. Write several sentences that explain how you constructed your graph, discussing when you expect \(v(t)\) to be zero, positive, negative, relatively large, and relatively small. d. Is there a connection between the two graphs that you can describe? What can you say about the velocity graph when the height function is increasing? decreasing? Make as many observations as you can.

An object moving along a straight line path has a differentiable position function \(y=s(t) ;\) \(s(t)\) measures the object's position relative to the origin at time \(t\). It is known that at time \(t=9\) seconds, the object's position is \(s(9)=4\) feet (i.e., 4 feet to the right of the origin). Furthermore, the object's instantaneous velocity at \(t=9\) is -1.2 feet per second, and its acceleration at the same instant is 0.08 feet per second per second. a. Use local linearity to estimate the position of the object at \(t=9.34\). b. Is your estimate likely too large or too small? Why? c. In everyday language, describe the behavior of the moving object at \(t=9 .\) Is it moving toward the origin or away from it? Is its velocity increasing or decreasing?
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