1.2E: Rates of Change (Exercises)
Section 1.2 Exercises
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The table below gives the annual sales (in millions of dollars) of a product each year from 1998 to 2006.
- What was the average rate of change of annual sales between 1998 and 2000?
- What was the average rate of change of annual sales between 2000 and 2002?
- What was the average rate of change of annual sales between 2002 and 2004?
- What was the average rate of change of annual sales between 2004 and 2006?
- Describe what is happening to the rate of change in sales over time.
| year | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 |
| sales | 201 | 219 | 233 | 243 | 249 | 251 | 249 | 243 | 233 |
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The table below gives the population of a town, in thousands each year from 2000 to 2008.
- What was the average rate of change of population between 2000 and 2002?
- What was the average rate of change of population between 2002 and 2004?
- What was the average rate of change of population between 2004 and 2006?
- What was the average rate of change of population between 2006 and 2008?
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Describe what is happening to the rate of change in sales over time.
year 2000 2001 2002 2003 2004 2005 2006 2007 2008 population 87 84 83 80 77 76 75 78 81
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The value of a company \(t\) days since January 1 is given in the table below.
- What was the average rate of change in stock value from \(t=0\) to \(t=10\)?
- What was the average rate of change in stock value from \(t=10\) to \(t=20\)?
- What was the average rate of change in stock value from \(t=20\) to \(t=40\)?
- Describe what is happening to the stock price over time
| \(t\) days since Jan. 1 | 0 | 10 | 20 | 30 | 40 |
| \(V\), Stock Value in $ | 123 | 144 | 161 | 174 | 185 |
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Based on the graph shown, estimate the average rate of change on each interval
- from \(x = 1\) to \(x = 4\).
- from \(x = 2\) to \(x = 5\).
- Find the average rate of change of each function on the interval specified of \(f(x)=x^{2}\) on [1, 5].
- Find the average rate of change of each function on the interval specified.\(q(x)=x^{3}\) on [-4, 2]
- Find the average rate of change of each function on the interval specified.\(g(x)=3x^{3} -1\) on [-3, 3]
- Find the average rate of change of each function on the interval specified.\(h(x)=5 - 2x^{2}\) on [-2, 4]
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Let \(f(x)= 4x^{2} -7\).
- Evaluate and simplify the difference quotient \(\dfrac{f(x)-f(a)}{x-a}\)
- Use the difference quotient to find the average rate of change of \(f\) on [1,4].
- Use the difference quotient to find the average rate of change of \(f\) on [4,8].
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Let \(g(x)= 2x^{2} -9\).
- Evaluate and simplify the difference quotient \( \dfrac{g(a+h)-g(a)}{h} \)
- Use the difference quotient to find the average rate of change of \(g\) on [2,5].
- Use the difference quotient to find the average rate of change of \(g\) on [5,9].
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Let \(k(x)= x^2+3x + 4\).
- Evaluate and simplify the difference quotient \( \dfrac{k(a+h)-k(a)}{h} \)
- Use the difference quotient to find the average rate of change of \(k\) on [2,5].
- Use the difference quotient to find the average rate of change of \(k\) on [5,9].
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Let \(f(x)= 4x^2 - 2x+1\).
- Evaluate and simplify the difference quotient \( \dfrac{f(a+h)-f(a)}{h} \)
- Use the difference quotient to find the average rate of change of \(f\) on [-2,1].
- Use the difference quotient to find the average rate of change of \(f\) on [1,6].
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Let \(g(t)=\dfrac{1}{t + 4}\).
- Evaluate and simplify the difference quotient \( \dfrac{g(x)-g(a)}{x-a} \)
- Use the difference quotient to find the average rate of change of \(g\) on [1,5].
- Use the difference quotient to find the average rate of change of \(g\) on [5,12].
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Suppose the profit of a company (in millions) that produces \(t\) million units of a product can be modeled by the function \(P(t)=10t-t^2\).
- Evaluate and simplify the difference quotient for \(P(t)\) to find a formula for the average rate of change in profit relative to production. Use either from of the difference quotient \( \dfrac{P(a+h)-P(a)}{h} \) or \( \dfrac{P(x)-P(a)}{x-a} \).
- Use the difference quotient to find the average rate of change in profit if production increased from 1 million units to 2 million units.
- Use the difference quotient to find the average rate of change in profit if production increased from 2 million units to 3 million units.
- Use the difference quotient to find the average rate of change in profit if production increased from 3 million units to 4 million units.
- Use your results in parts (b)-(d) to describe what it happening to the profit with this increased production.-
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An automobile is driven down a straight highway. It position (in feet) after t seconds can be modeled by the function \(s(t)=4.5t^2\).
- Evaluate and simplify the difference quotient for \(P(t)\) to find a formula for the average velocity (the average rate of change in position relative to time). Use either from of the difference quotient \( \dfrac{s(a+h)-s(a)}{h} \) or \( \dfrac{s(x)-s(a)}{x-a} \).
- Use the difference quotient to find the average rate of change in position on the interval [6,7].
- Use the difference quotient to find the average rate of change in position on the interval [6,6.1].
- Use the difference quotient to find the average rate of change in position on the interval [6,6.01].
- Use the difference quotient to find the average rate of change in position on the interval [6,6.001].
- What do your results in parts (b)-(e) tell you about the velocity at the instant \(t=6\) seconds?
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The pressure of 10 moles of an ideal gas is given by the formula \(P=f(V)=\dfrac{240}{V\)} at the appropriate temperature. Find an interpret each of the following quantities.
- \(f(8)\)
- \(f(8)-f(5)\)
- \(\dfrac{f(8)-f(5)}{8-5} \)
- Answer
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1. (a) 16 million dollars per year (b) 8 million dollars per year (c) 0 , no change in sales (d) -8 million dollars per year (e) The increase in annual sales is getting smaller and smaller over time before eventually leveling off and then decreasing.
3. (a) $2.1 dollars per day (b) $1.7 dollars per day (c) $1.3 dollars per day (d) $1.15 dollars per day (e) The stock value is increasing at a smaller and smaller rate over time.
5. 6
7. 27
9. (a) \(\dfrac{\Delta y }{\Delta x } = 4(x+a)\) (b) 20 (c) 48
11. (a) \(\dfrac{\Delta y }{\Delta x } = 2a+h+3\) (b) 10 (c) 17
13. (a) \(\dfrac{\Delta y }{\Delta x } = \dfrac{-1}{(x+4)(a+4) }\) (b) \(\dfrac{-1}{45}\) (c) \(\dfrac{-1}{144}\)
15. (a) \(\dfrac{\Delta s }{\Delta t } = 4.5(x+a)\) (b) 58.5 \(\dfrac{\text{ft}}{\text{s}}\) (c) 54.45 \(\dfrac{\text{ft}}{\text{s}}\) (d) 54.045 \(\dfrac{\text{ft}}{\text{s}}\) (e) 54.0045 \(\dfrac{\text{ft}}{\text{s}}\) (f) The velocity at the instant \(t= 6\) seconds is 54 \(\dfrac{\text{ft}}{\text{s}}\).