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Chapter 4: Problem 18

The derivative of \(f(t)\) is given by \(f^{\prime}(t)=t^{3}-6 t^{2}+8 t\) for \(0 \leq t \leq 5\). Graph \(f^{\prime}(t)\), and describe how the function \(f(t)\) changes over the interval \(t=0\) to \(t=5 .\) When is \(f(t)\) increasing and when is it decreasing? Where does \(f(t)\) have a local maximum and where does it have a local minimum?

Short Answer

Expert verified
\(f(t)\) is increasing on \(0 < t < 2\) and \(4 < t < 5\); decreasing on \(2 < t < 4\). Local max at \(t = 2\) and local min at \(t = 4\).

Step by step solution

01

Understand Derivative and Critical Points

First, we need to find the critical points of the function by setting the derivative equal to zero: \[ t^3 - 6t^2 + 8t = 0. \] This equation can be factored as: \[ t(t^2 - 6t + 8) = 0. \]Further factoring yields:\[ t(t-2)(t-4) = 0. \]Thus, the critical points are \( t = 0, 2, \) and \( 4. \)
02

Determine Intervals of Increase and Decrease

To know where \( f(t) \) is increasing or decreasing, we check the sign of \( f'(t) \) between the critical points.- For \( 0 < t < 2 \), choose \( t = 1 \): \[ f'(1) = 1^3 - 6 imes 1^2 + 8 imes 1 = 3 \] (positive, so increasing).- For \( 2 < t < 4 \), choose \( t = 3 \): \[ f'(3) = 3^3 - 6 imes 3^2 + 8 imes 3 = -3 \] (negative, so decreasing).- For \( 4 < t < 5 \), choose \( t = 4.5 \): \[ f'(4.5) = (4.5)^3 - 6 imes (4.5)^2 + 8 imes 4.5 = 2.625 \] (positive, so increasing).
03

Identify Local Maxima and Minima

From the sign changes:- At \( t = 2 \), \( f'(t) \) changes from positive to negative, indicating a local maximum.- At \( t = 4 \), \( f'(t) \) changes from negative to positive, indicating a local minimum.
04

Sketch the Graph

Sketch or simply understand the graph of \( f'(t) = t^3 - 6t^2 + 8t \):- The x-axis intercepts at points \( t = 0, 2, \) and \( 4 \).- Increasing on intervals \( (0, 2) \) and \( (4, 5) \).- Decreasing on interval \( (2, 4) \).This helps to visually confirm where the function \( f(t) \) is increasing, decreasing, and has local extrema.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points
To find critical points, we set the derivative of the function equal to zero. These points occur where there could be a peak, valley, or a flat spot on the graph. For the function with derivative \(f^{\prime}(t) = t^{3}-6t^{2}+8t\), we start by solving \(t^{3} - 6t^{2} + 8t = 0\). This equation can be factored as \(t(t-2)(t-4) = 0\), giving us critical points at \(t = 0, 2, \text{ and } 4\). Critical points are crucial because they help us determine the behavior of the function. Each critical point is a candidate for a local maximum or minimum, and understanding them aids in analyzing the function's overall dynamics.
Intervals of Increase and Decrease
Once we've identified the critical points, we check the intervals between them to see if the function is increasing or decreasing.- For \(0 < t < 2\), we test a point within this interval like \(t = 1\). Since \(f'(1) > 0\), the function is increasing.- For \(2 < t < 4\), we choose \(t = 3\) and find that \(f'(3) < 0\), so here the function is decreasing.- Lastly, for \(4 < t < 5\), testing \(t = 4.5\) shows \(f'(4.5) > 0\), indicating the function is increasing again.Understand that the sign of the derivative \(f'(t)\) tells us if the original function \(f(t)\) is rising or falling in these segments. Positive derivative means increasing, while negative means decreasing.
Local Maximum and Minimum
Identifying local maxima and minima involves looking at the transitions of \(f'(t)\) between positive and negative signs. - At \(t = 2\), \(f'(t)\) changes from positive to negative. This change tells us \(f(t)\) reaches a peak here, signaling a local maximum.- Conversely, at \(t = 4\), \(f'(t)\) shifts from negative to positive, suggesting \(f(t)\) has a dip, which is a local minimum.These local extrema provide valuable insights into the function's graph. It's where the function behaves contrary to its nearby intervals and thus stands as significant markers to sketching or understanding the function's behavior over the given domain.

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Most popular questions from this chapter

A curve representing the total number of people, \(P\), infected with a virus often has the shape of a logistic curve of the form $$ P=\frac{L}{1+C e^{-k t}} $$ with time \(t\) in weeks. Suppose that 10 people originally have the virus and that in the early stages the number of people infected is increasing approximately exponentially, with a continuous growth rate of \(1.78 .\) It is estimated that, in the long run, approximately 5000 people will become infected. (a) What should we use for the parameters \(k\) and \(L\) ? (b) Use the fact that when \(t=0\), we have \(P=10\), to find \(C .\) (c) Now that you have estimated \(L, k\), and \(C\), what is the logistic function you are using to model the data? Graph this function. (d) Estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the value of \(P\) at this point?

The quantity of a drug in the bloodstream \(t\) hours after a tablet is swallowed is given, in \(\mathrm{mg}\), by $$q(t)=20\left(e^{-t}-e^{-2 t}\right)$$ (a) How much of the drug is in the bloodstream at time \(t=0 ?\) (b) When is the maximum quantity of drug in the bloodstream? What is that maximum? (c) In the long run, what happens to the quantity?

If \(R\) is percent of maximum response and \(x\) is dose in \(\mathrm{mg}\), the dose-response curve for a drug is given by $$ R=\frac{100}{1+100 e^{-0.1 x}} $$ (a) Graph this function. (b) What dose corresponds to a response of \(50 \%\) of the maximum? This is the inflection point, at which the response is increasing the fastest. (c) For this drug, the minimum desired response is \(20 \%\) and the maximum safe response is \(70 \%\). What range of doses is both safe and effective for this drug?

For the functions in problems, do the following: (a) Find \(f^{\prime}\) and \(f^{\prime \prime}\). (b) Find the critical points of \(f\). (c) Find any inflection points of \(f\). (d) Evaluate \(f\) at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of \(f\) in the interval. (e) Graph \(f\). $$ f(x)=x+\sin x \quad(0 \leq x \leq 2 \pi) $$

A business sells an item at a constant rate of \(r\) units per month. It reorders in batches of \(q\) units, at a cost of \(a+b q\) dollars per order. Storage costs are \(k\) dollars per item per month, and, on average, \(q / 2\) items are in storage, waiting to be sold. [Assume \(r, a, b, k\) are positive constants.] (a) How often does the business reorder? (b) What is the average monthly cost of reordering? (c) What is the total monthly cost, \(C\) of ordering and storage? (d) Obtain Wilson's lot size formula, the optimal batch size which minimizes cost.
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