- A car’s velocity follows the equation [latex]v(t) = 10t - t^2[/latex] feet per second from [latex]t = 0[/latex] to [latex]t = 10[/latex]. How far does the car travel during this time period?
[latex]\approx 166.7[/latex] feetans
- A car’s velocity follows the equation [latex]v(t) = 10 - \sqrt{t}[/latex] from [latex]t = 0[/latex] to [latex]t = 100[/latex]. How far does the car travel from [latex]t = 0[/latex] to [latex]t = 100[/latex]?
[latex]\approx 333.33[/latex] unitsans
- A car’s acceleration follows the equation [latex]a(t) = t[/latex] from [latex]t = 0[/latex] to [latex]t = 10[/latex]. Recall that acceleration is the derivative of velocity.
- Find a function [latex]v(t)[/latex] for the velocity at time [latex]t[/latex].
[latex]v(t) =\frac{1}{2} t^2[/latex] (you could also add any constant to this and still have a valid answer.)ans
- How far does the car travel from [latex]t = 0[/latex] to [latex]t = 10[/latex]?
Need to compute [latex]\int_{0^10} \frac{1}{2} t^2dt \approx 166.7[/latex] units.ans
- Find a function [latex]v(t)[/latex] for the velocity at time [latex]t[/latex].
- An employee’s wages start at $10,000 a year and quickly increase after that at a rate of [latex]0.04[/latex] per year, continuously implemented. Thus, at year [latex]t[/latex], the employee makes
[latex]10000e^{0.0277t} [/latex]
dollars per year.
- How much does the employee make per year at year [latex]5[/latex]?
[latex]\$11485.5[/latex]ans
- How much total does the employee make in the first five years?
[latex]\int_0^5 10000e^{0.0277t}dt \approx \$53628[/latex]ans
- How much does the employee make per year at year [latex]5[/latex]?
- Water drains from a tub at a rate of [latex]\sqrt{50 - 2t}[/latex] gallons per minute, with [latex]t[/latex] measured in minutes.
- How long does it take for the rate to drop to zero?
[latex]t = 25[/latex]ans
- How much total water has been lost at this point?
[latex]\int_0^{25} \sqrt{50-2t} dt \approx 69.28[/latex]ans
- How long does it take for the rate to drop to zero?
- A biologist models elk growth rate as [latex]G(t) = 5 e^{0.02 t}[/latex] measured in elk per year.
- How fast is the elk growth rate changing at [latex]t = 10[/latex]?
[latex]0.122[/latex] elk per year per yearans
- How many elk were born in the first 20 years of this model?
[latex]\int_0^{20} 5 e^{0.02 t} \approx 123[/latex]ans
- Do a sensitivity analysis. Given a small change in [latex]5[/latex], how does that affect the answer to part (a)? Given a small change in [latex]0.02[/latex], how does that affect part (a)?
- How fast is the elk growth rate changing at [latex]t = 10[/latex]?
- Let [latex]G(t)[/latex] be the rate at which GDP is growing measured in dollars per day. Match the symbols [latex]G'(t)[/latex], [latex]G(t)[/latex] and [latex]\int_0^t G(t)dt[/latex] to the following statements.
- This measures the rate that GDP growth is speeding up or slowing down.
[latex]G'(t)[/latex]ans
- This measures how much GDP has increased since the beginning of the year.
[latex]\int_0^t G(t)dt[/latex]ans
- This measure how quickly GDP is increasing.
[latex]G(t)[/latex]ans
- This measures the rate that GDP growth is speeding up or slowing down.
- The Greenland ice sheet is losing ice. It is estimated that it is losing ice at a rate of [latex]f(t) = -0.5t^2-150[/latex] gigatonnes per year, with [latex]t[/latex] measured in years, and [latex]t = 0[/latex] representing [latex]2010[/latex]. How many gigatonnes of ice will the ice sheet lose from [latex]2015[/latex] to [latex]2025[/latex]?
[latex]\int_5^{15} -0.5t^2-150dt \approx -2040[/latex] gigatonnes.ans
- Let [latex]C(t)[/latex] be the crime rate in the city of Gotham, with [latex]C(t)[/latex] measured in crimes per day, and [latex]t[/latex] measured in days. Match [latex]C(t)[/latex], [latex]C'(t)[/latex], and [latex]\int C(t) dt[/latex] to the following.
- This function would tell you how many crimes are committed over the last 90 days.
[latex]\int C(t)dt[/latex]ans
- This function would tell you how many crimes per day were being committed 90 days ago.
[latex]C(t)[/latex]ans
- This function will tell you how quickly the crime rate was increasing or decreasing 90 days ago.
[latex]C'(t)[/latex]ans
- This function would tell you how many crimes are committed over the last 90 days.
- When blasting off from the earth into space, a rocket uses fuel at a rate of [latex]f(t) = 5 + 100e^{-0.01t}[/latex], where [latex]t[/latex] is measured in seconds and [latex]f(t)[/latex] is measured in gallons per second.
- How many gallons are used in a four-minute flight starting at [latex]t = 0[/latex].
[latex]\int_0^4 5 + 100e^{-.01t} \approx 412[/latex]ans
- How many gallons are used in a two-minute flight starting at [latex]t = 0[/latex]?
[latex]\int_0^2 5 + 100e^{-0.01t} \approx 208[/latex]ans
- Should your answer for (b) be exactly half of the answer for part (a)? Why or why not?
No, since rockets don’t use fuel at a constant rate.ans
- How many gallons are used in a four-minute flight starting at [latex]t = 0[/latex].
- The amount of sun power that is available to a flower is given by [latex]S(t) = 2.5 \sin\left( \frac{\pi}{12} t \right) + 2.5[/latex] kilojoules per hour. The flower can absorb energy at [latex]5%[/latex] efficiency, meaning it can use or store about [latex]5%[/latex] of the available sunlight energy. How much energy (in kilojoules) does the flower absorb in a [latex]48[/latex]-hour period?
[latex]6[/latex] kilojoulesans
- Submarine Navigation
Nuclear submarines spend months underwater with no access to GPS or similar navigation techniques. Instead, they use a “dead reckoning” approach where accelerometers are used to keep track of how fast they are moving, from which their position can be determined. A submarine starts not moving at all. Given the following list of accelerations, estimate how far the submarine has gone.[latex]\begin{array}{cc} \text{Day} & \text{Average Acceleration in miles / day$^2$} \\ 0 & 220 \\ 1 & 135 \\ 2 & -150 \\ 3 & 0 \\ 4 & 0 \\ 5 & 200 \\ 6 & 0 \\ 7 & -405 \\ 8 & 0 \end{array}[/latex]