3.6: Derivatives as Rates of Change
- Page ID
- 144210
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- The position of a drone flying along a straight line is given by the function \(x(t) = 2t^3 - 6t\), where \(t\) is in seconds and \(x(t)\) is in meters.
- What is the initial velocity of the drone?
- What is the drone's velocity after \(t = 1\) second?
- What is the initial acceleration of the drone?
- What is the drone's acceleration after \(t = 1\) second?
- What is the initial velocity of the drone?
- A ball is dropped from the top of a building 256 feet tall. The ball's height in feet above the ground \(t\) seconds after it is released is given by \(y(t) = -16t^2 + 256\) .
- How long does it take for the ball to hit the ground?
- What is the velocity of the ball at the moment when it is released?
- What is the velocity of the ball when it hits the ground?
- What is the acceleration of the ball at the moment when it is released?
- What is the acceleration of the ball when it hits the ground?
- How long does it take for the ball to hit the ground?
- The position in miles of a freight train after \(t\) hours is given by the function \(s(t) = t^3 - 4t\), where east is the positive direction.
- Determine the direction and speed of the train when \(s(t) = 0\).
- Determine the direction and speed of the train when \(a(t) = 0\).
- Determine the direction and speed of the train when \(s(t) = 0\).
- A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town's population. The study found that the town's population can be modeled by the function \(P(t) = -\dfrac{1}{3}t^3 + 64t + 3000\), where \(t\) is measured in years.
- Find \(P'(1)\), \(P'(2)\), \(P'(3)\), and \(P'(4)\). Interpret what the results mean for the town's population.
- Find \(P''(1)\), \(P''(2)\), \(P''(3)\), and \(P''(4)\). Interpret what the results mean for the town's population.
- Find \(P'(1)\), \(P'(2)\), \(P'(3)\), and \(P'(4)\). Interpret what the results mean for the town's population.
- A culture of bacteria grows in number according to the function \(N(t) = 3000\left(1 + \dfrac{4t}{t^2 + 100}\right)\), where \(t\) is measured in hours.
- Find the rate of change of the number of bacteria, \(N'(t)\).
- Using a calculator, find \(N'(0)\), \(N'(10)\), \(N'(20)\), and \(N'(30)\). Interpret what the results imply about the bacteria population.
- Find \(N''(t)\).
- Using a calculator, find \(N''(0)\), \(N''(10)\), \(N''(20)\), and \(N''(30)\). Interpret what the results imply about the bacteria population.
- Find the rate of change of the number of bacteria, \(N'(t)\).
- The cost in dollars for a company to manufacture \(x\) food processors is given by \(C(x) = 200 + \dfrac{7}{x} + \dfrac{x}{27}\).
- Find the marginal cost of manufacturing 12 food processors. Using your result, estimate the cost of producing the thirteenth food processor.
- Find the actual cost of manufacturing the thirteenth food processor.
- Find the marginal cost of manufacturing 12 food processors. Using your result, estimate the cost of producing the thirteenth food processor.
- The total revenue (in dollars) that a company makes for manufacturing and selling \(x\) digital clock radios is \(R(x) = 10x - 0.001x^2\).
- Find the marginal revenue function.
- Calculate the marginal revenue for \(x = 2000\) and \(x = 5000\). Interpret your results.
- Find the marginal revenue function.