Homework:  Second Derivatives and Interpreting the Derivative

Tyler Seacrest

38 Homework: Second Derivatives and Interpreting the Derivative

Given each $f (t)$ , describe in one sentence the meaning of $f^{'} (t)$ .
1. Let $f (t)$ be the distance (in miles) an astronaut is from the surface of the earth as he blasts off towards space. Here $t$ is measured in hours.
  
  $f^{'} (t)$ is the speed of the astronaut in miles per hour
  
  ans
2. Let $f (t)$ be the number of gallons of diesel gasoline in the tank of a truck, with $t$ measured in hours.
  
  $f^{'} (t)$ is how fast fuel is being burned, in gallons per minute. It could also represent how fast the fuel is being filled up at a gas station.
  
  ans
3. Let $f (t)$ be the concentration of NaCl in parts per million within the cytoplasm of a cell. Here, $t$ is measured in minutes.
  
  $f^{'} (t)$ is the rate the concentration of NaCl is increasing in parts per million per minute.
  
  ans
4. Let $f (t)$ be the speed of a runner (in feet per second), and let $t$ be measured in seconds.
  
  $f^{'} (t)$ is the acceleration of the runner, or the rate at which the runner’s speed is increasing in (feet per second) per second.
  
  ans
5. Let $f (t)$ be the rate (in dollars per hour) that you are paid, where $t$ is measured in months.
  
  $f^{'} (t)$ is like how fast are you getting raises, measured in (dollars per hour) per month.
  
  ans
For each of the functions below, compute the derivative twice. That is, compute $f^{'} (t)$ , then take the derivative of $f^{'} (t)$ to find $f^{″} (t)$ .
1. $f (t) = 5 t^{2} - 6 t + 2$
  
  $10 t - 6$ , $10$
  
  ans
2. $f (t) = t^{3} + e^{t} - \sqrt{t}$ .
  
  $3 t^{2} + e^{t} - 1 / 2 t^{- 1 / 2}$ , $6 t + e^{t} + 1 / 4 t^{- 3 / 2}$
  
  ans
3. $f (t) = \frac{1}{t}$
  
  $- t^{- 2}, 2 t^{- 3}$
  
  ans
4. $f (t) = \sin (t)$
  
  $\cos (t)$ , $(- \sin (t))$
  
  ans
5. $f (t) = (\sqrt{t} + 1)^{2}$
  
  $1 + t^{- 1 / 2}$ , $- \frac{1}{2} t^{- 3 / 2}$
  
  ans
For each graph, circle any inflection points (if any). Label each region between the inflection points has having either a positive or negative second derivative.
ans
A river bank is eroding. Let $f (t)$ be the metric tonnes of soil and rock material on day $t$ . Suppose $f (t) = 0.05 t^{2} - 3 t + 125$ (this model is valid from $t = 0$ to $t = 30$ ) .
1. What is $f^{'} (t)$ measuring? What are the correct units?
  
  How quickly soil and rock are being lost due to erosion in metric tonnes per day.
  
  ans
2. Sketch the derivative.
3. Find the derivative
  
  $0.1 x - 3$
  
  ans
4. How quickly is material being lost on day $t = 5$ ? How quickly is material being lost on day $t = 25$ ?
  
  $2.5$ metric tons per day on day $5$ , $0.5$ metric tons per day on day $25$ .
  
  ans
The stock price for Math Nerds, Inc, over the course of an 8 hour trading day ( $t = 0$ to $t = 8$ ) is modeled by $p (t) = - 0.3 t^{3} + 3 t^{2} - 4 t + 17$ ( $p (t)$ is measured in dollars).
1. What is $p^{'} (t)$ measuring? What is $p^{″} (t)$ measuring? State the correct units for each.
  
  $p^{'} (t)$ is the rate of change of the stock price in dollars per hour. $p^{″} (t)$ is how quickly the stock price rate is speeding up or slowing down, measured in dollars per hour $^{2}$
  
  ans
2. Sketch the graph of $p^{'} (t)$ and $p^{″} (t))$ .
3. Compute $p^{'} (t)$ and $p^{″} (t)$ .
  
  $p^{'} (t) = - 0.9 t^{2} + 6 t - 4$ , $p^{″} (t) = - 1.8 t + 6$ .
  
  ans
4. How quickly is the stock gaining in price at $t = 4$ ? How quickly is it losing value at $t = 7$ ?
  
  Gaining at 5.6 dollars per hour at $t = 4$ , but losing at a rate of $- 6.1$ dollars per hour at $t = 7$
  
  ans
5. At $t = 4$ , the stock price is clearly growing. But is growth speeding up or slowing down? How can you use the formula for $p (t)$ , $p^{'} (t)$ , or $p^{″} (t)$ to find out?
  
  We can plug $t = 4$ into $p^{″} (t)$ , to get an answer of $- 1.2$ dollars per hour per hour, which means growth is slowing down.
  
  ans
A tank has $f (t)$ liters of water at time $t$ measured in minutes, where $f (t) = - 2.2 t^{3} + 27 t^{2} - 120 t + 500$ (this model is valid $t = 0$ to $t = 9$ ).
1. Sketch the graphs of $f^{'} (t)$ and $f^{″} (t)$ .
2. What is $f^{'} (t)$ measuring? What is the meaning of $f^{″} (t)$ ? Give the correct units.
  
  Imagine water is leaking out of a hole. $f^{'} (t)$ is indirectly measuring the size of that hole, since it is measuring how quickly water is being lost in liters per minute. $f^{″} (t)$ would relate to how quickly the hole is opening or closing, thus measuring how fast the rate is changing in liters per minute per minute.
  
  ans

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38 Homework: Second Derivatives and Interpreting the Derivative

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