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In part <xrefref="PA-1-1"text="type-global"/>, you explored the velocity of a ball at various times. Answer the following questions to explore other rates of change.
For each of the following scenarios, specify the independent (input) and dependent (output) variables, along with possible units for both. Then give units for the rate of change.
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The population of a city changes over time.
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Independent: <response />
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Dependent: <response />
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Rate of change: <response />
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As a weather balloon rises, the temperature of the air inside (and outside) the balloon decreases.
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Independent: <response />
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Dependent: <response />
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Rate of change: <response />
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For a given trip (i.e. a set distance) as the speed of the car increases, the travel time decreases.
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Independent: <response />
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Dependent: <response />
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Rate of change: <response />
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As the population of a city decreases, the tax revenue of the city also decreases.
Each of the functions on the left below could be described as having a specific algebraic structure as noted on the right. Match each function with its corresponding structure. It is possible that not all structures are used; and it is possible that not all functions have a description for its algebraic structure.
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In <xrefref="PA-2-1">Preview Activity</xref>, we learned how to use the limit definition of the derivative to find a formula for <m>f'(x)</m> for a collection of power functions. A different way to think about derivative formulas for certain kinds of functions is to use the perspective that the derivative measures the slope of the tangent line to a given function.
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Consider the function <m>f(x) = x</m>, and observe that this function is linear. What is the value of the slope of <m>f(x) = x</m> at any given point on the graph? What should be the formula for <m>f'(x)</m>?
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For the linear function <m>g(x)=2x+1</m>, what is the value of the slope of <m>g</m> at any point on the graph? What should be the formula for <m>g'(x)</m>?
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If <m>h(x)=7-\frac{2}{3}x</m>, what do you expect will be the formula for <m>h'(x)</m>?
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Suppose that <m>p(x) = 3x - 5</m> and <m>q(x) = 4x + 8</m>. If <m>r(x) = p(x) + q(x)</m>, what do you think will be the formula for <m>r'(x)</m>? Why?
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Suppose that <m>p(x) = 3x - 5</m>. If <m>s(x) = -4 \cdot p(x)</m>, what do you think will be the formula for <m>s'(x)</m>? Why?
In <xrefref="PA-2-2">Preview Activity</xref>, we reasoned graphically to sketch the derivative of <m>g(x) = 2^x</m>. The following questions remind us of some key ideas related to sketching the graph of the derivative, given a graph of the original function.
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Let <m>q(x) = 2 - (x-1)^2</m>, and use a graphing utility to plot this quadratic function. At what value(s) of <m>x</m> is <m>q'(x) = 0</m>? For what values of <m>x</m> is <m>q'(x)</m> positive? For what values of <m>x</m> is <m>q'(x)</m> negative?
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Let <m>f(x) = \sin(x)</m>, and use a graphing utility to plot this trigonometric function. State 3 values of <m>x</m> for which <m>f'(x) = 0</m>. What is one interval values of <m>x</m> on which <m>f'(x)</m> positive? What is one interval of values of <m>x</m> on which <m>f'(x)</m> negative?
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Let <m>g(x) = \cos(x)</m>, and use a graphing utility to plot this trigonometric function. State 3 values of <m>x</m> for which <m>g'(x) = 0</m>. What is one interval values of <m>x</m> on which <m>x</m> is <m>g'(x)</m> positive? What is one interval values of <m>x</m> on which <m>g'(x)</m> is negative?
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