1.4E: More on Using Functions to Describe Relationships (Exercises)
Section 1.4 Exercises
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For the function graphed below,
- Estimate the intervals on which the function is increasing and decreasing.
- Determine on which intervals the rate of increase is getting larger in size.
- Determine on which intervals the rate of increase is getting smaller in size.
- Determine on which intervals the rate of decrease is getting larger in size.
- Determine on which intervals the rate of decrease is getting smaller in size.
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For the function graphed below,
- Estimate the intervals on which the function is increasing and decreasing.
- Determine on which intervals the rate of increase is getting larger in size.
- Determine on which intervals the rate of increase is getting smaller in size.
- Determine on which intervals the rate of decrease is getting larger in size.
- Determine on which intervals the rate of decrease is getting smaller in size.
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For the function graphed below,
- Estimate the intervals on which the function is increasing and decreasing.
- Determine on which intervals the rate of increase is getting larger in size.
- Determine on which intervals the rate of increase is getting smaller in size.
- Determine on which intervals the rate of decrease is getting larger in size.
- Determine on which intervals the rate of decrease is getting smaller in size.
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For the function graphed below,
- Estimate the intervals on which the function is increasing and decreasing.
- Determine on which intervals the rate of increase is getting larger in size.
- Determine on which intervals the rate of increase is getting smaller in size.
- Determine on which intervals the rate of decrease is getting larger in size.
- Determine on which intervals the rate of decrease is getting smaller in size.
5. For each table below, Determine whether the table represents a function that is increasing or decreasing. Then determine whether the rate of increase or decrease is getting larger or smaller in size.
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\(x\) 1 2 3 4 5 \(f(x)\) 2 4 8 16 32 -
\(x\) 1 2 3 4 5 \(g(x)\) 90 80 75 72 71 -
\(x\) 1 2 3 4 5 \(h(x)\) 300 290 270 240 200 -
\(x\) 1 2 3 4 5 \(k(x)\) 0 15 25 32 35 -
\(x\) 1 2 3 4 5 \(L(x)\) -10 -25 -37 -47 -54 -
\(x\) 1 2 3 4 5 \(m(x)\) -200 -190 -160 -100 0 -
\(x\) 1 2 3 4 5 \(n(x)\) -100 -50 -25 -10 0 -
\(x\) 1 2 3 4 5 \(p(x)\) -50 -100 -200 -400 -900
6. A person pours a hot cup of tea into a cup with a temperature of 170 \(^{\circ} F\). The temperature of the cup of tea decreases over time. Initially, the temperature drops quickly then drops more slowly over time. Draw a graph which represents this scenario.
7. A company's monthly revenue is $1.2 million in Jan. 2022. Due to a recent product launch, the company's revenue increasing dramatically at first, then starts to increase more slowly over time as the popularity and newness of the product fades. Draw a graph which represents this scenario.
8. Identify the domain and range of each function. Write your result using interval notation or an inequality.
a) b)
9. Identify the domain and range of each function. Write your result using interval notation or an inequality.
a) b)
10. Suppose that you are holding your toy submarine under the water. You release it and it begins to ascend. The graph models the depth of the submarine as a function of time, stopping once the sub surfaces. Draw a graph that models this situations. What is the domain and range of the function in the graph?
11. Find the domain of each function
a) \(f(x)=3x+1\) b) \(f(x)=3\sqrt{x-2}\) c) \(f(x)=3-\sqrt{6-2x}\) d) \(f(x)=\dfrac{9}{x - 6}\)
12. Find the domain of each function
a) \(f(x)=3x+1\) b) \(f(x)=5\sqrt{x+3}\) b) \(f(x)=5-\sqrt{10-2x}\) c) 12. \(f(x)=\dfrac{6}{x - 8}\)
13. Find the domain of each function
a) \(f(x)=\dfrac{3x+1}{4x+2}\) b) \(f(x)=\dfrac{\sqrt{x+4} }{x-4}\) c) \(f(x)=\dfrac{x -3}{x^{2} + 9x -22}\)
14. Find the domain of each function
a) \(f(x)=\dfrac{5x+3}{4x-1}\) b) \(f(x)=\dfrac{\sqrt{x+5} }{x-6}\) c) \(f(x)=\dfrac{x -8}{x^{2} + 8x -9}\)
15. The discharge of the San Joaquin River out of Millerton Lake dam in \(\dfrac{\text{ft}^{3}}{s} \) \(t\) days since February 1,2024 is modeled by the equation \(D=f(t)=571-0.288t \) from exercise 1.3.5.
a) Identify the mathematical restrictions on the input \(t\).
b) Identify the restrictions on the input \(t\) so that \(t\) makes sense as a time since February 1, 2024.
c) Identify the restrictions on the input \(t\) so that \(D\) makes sense as a discharge from the San Joaquin River.
d) Find the domain of the application.
16. A balloon is filled with 40 \(\text{in}^{3}\) of air. Due to a small hole, the air is released from a balloon at a rate of 1.2 \(\dfrac{\text{in}^{3}{s} \). The volume after t seconds is given by the formula \(V=f(t)=40 - 1.2t\). Find the domain of the application.
17. Suppose the revenue (or total sales) of a company (in millions) that produces \(x\) thousand units of a product can be modeled by the function \(R=R(x)=300x-2x^2\).
a) Identify the mathematical restrictions on the input \(x \).
b) Identify the restrictions on the input \(x \) so that \(x \) makes sense as the thousand of units of a product produced.
c) Identify the restrictions on the input \(x \) so that \(R \) makes sense as a revenue.
d) Find the domain of the application.
18. An open box is to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Let \(x\) represent the side length of the cut out.
a) The function for the length of the side of the box is \(L=L(x)=16-2x\). Find the domain of the application.
b) The function for the width of the side of the box is \(W=W(x)=16-2x\). Find the domain of the application.
c) The function for the height of the side of the box is \(H=H(x)=x\). Find the domain of the application.
d) Find a formula for the volume of the box \(V(x)\) in terms of \(x\) only. Then find the domain of the application.
- Answer
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1. a) Increasing on (-1.5,2) and decreasing on \( (-\infty,-1.5) \cup (2, \infty)\).
b) (-1.5,0.25) c) (0.25,2) d) \((2, \infty)\) e) \( (-\infty,-1.5)\)
3. a) Increasing on \( (-\infty,1) \cup (3, 4)\) and decreasing on \( (1,3) \cup (4, \infty)\).
b) (3,4) c) \( (-\infty,1)\) d) nowhere e) \( (1,3) \cup (4, \infty)\)
5. a) The function is increasing. The rate of increase is getting larger in size.
b) The function is decreasing. The rate of decrease is getting larger in size.
c) The function is decreasing. The rate of decrease is getting smaller in size.
d) The function is decreasing. The rate of decrease is getting larger in size.
7.
9. a) D: (2, 8] R: [6, 8) b) D: (4, 8] R: [2, 8)
11. a) \((-\infty, \infty)\) b) \([2, \infty)\) c) \((-\infty, 3]\) d) \((-\infty,6) \cup (6, \infty)\) or \(x \ne 6\)
13. a) \((-\infty, -\dfrac{1}{2}) \cup (-\dfrac{1}{2}, \infty)\) or \(x \ne -\dfrac{1}{2}\) b) \((-\infty, 4) \cup (4, \infty)\) or \(x \ne 4\) c) \((-\infty, -11) \cup (-11,2) \cup (2, \infty)\) or \(x \ne -11\) and \(x \ne 2\)
15. a) no restrictions b) \([0, \infty)\) c) \((-\infty, 1982]\) d) \([0,1982]\)
17. a) no restrictions b) \([0, \infty)\) c) \((0, 150]\) d) \([0,150]\)