Ask your homework questions to teachers and professors, meet other students, and be entered to win $600 or an Xbox Series X 🎉Join our Discord! ## Chapter 4 ## Applications of Differentiation ## Educators MP FP WZ + 4 more educators ### Problem 1 Explain the difference between an absolute minimum and a local minimum. FP Fahad P. Numerade Educator ### Problem 2 Suppose$ f $is a continuous function defined on a closed interval$ [a, b] $. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for$ f $? (b) What steps would you take to find those maximum and minimum values? FP Fahad P. Numerade Educator ### Problem 3 For each of the numbers$ a $,$ b $,$ c $,$ d $,$ r $, and$ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. FP Fahad P. Numerade Educator ### Problem 4 For each of the numbers$ a $,$ b $,$ c $,$ d $,$ r $, and$ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. Carson M. Numerade Educator ### Problem 5 Use the graph to state the absolute and local maximum and minimum values of the function. FP Fahad P. Numerade Educator ### Problem 6 Use the graph to state the absolute and local maximum and minimum values of the function. Carson M. Numerade Educator ### Problem 7 Sketch the graph of a function$ f $that is continuous on$ [1, 5] $and has the given properties. Absolute maximum at$ 5 $, absolute minimum at$ 2 $, local maximum at$ 3 $, local minima at$ 2 $and$ 4 $Chris T. Numerade Educator ### Problem 8 Sketch the graph of a function$ f $that is continuous on$ [1, 5] $and has the given properties. Absolute maximum at$ 4 $, absolute minimum at$ 5 $, local maximum at$ 2 $, local minimum at$ 3 $Chris T. Numerade Educator ### Problem 9 Sketch the graph of a function$ f $that is continuous on$ [1, 5] $and has the given properties. Absolute minimum at$ 3 $, absolute maximum at$ 4 $, local maximum at$ 2 $Chris T. Numerade Educator ### Problem 10 Sketch the graph of a function$ f $that is continuous on$ [1, 5] $and has the given properties. Absolute maximum at$ 2 $, absolute minimum at$ 5 $,$ 4 $is a critical number but there is no local maximum and minimum there. Chris T. Numerade Educator ### Problem 11 (a) Sketch the graph of a function that has a local maximum of$ 2 $and is differentiable at$ 2 $. (b) Sketch the graph of a function that has a local maximum of$ 2 $and is continuous but not differentiable at$ 2 $. (c) Sketch the graph of a function that has a local maximum of$ 2 $and is not continuous at$ 2 $. FP Fahad P. Numerade Educator ### Problem 12 (a) Sketch the graph of a function on$ [-1, 2] $that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on$ [-1, 2] $that has a local maximum but no absolute maximum. Chris T. Numerade Educator ### Problem 13 (a) Sketch the graph of a function on$ [-1, 2] $that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on$ [-1, 2] $that is discontinuous but has both an absolute maximum and an absolute minimum. Chris T. Numerade Educator ### Problem 14 (a) Sketch the graph of a function that has two local maximum, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers. Chris T. Numerade Educator ### Problem 15 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \frac{1}{2}(3x-1) $,$ x \leqslant 3 $Chris T. Numerade Educator ### Problem 16 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = 2 - \frac{1}{3}x $,$ x \geqslant -2 $Chris T. Numerade Educator ### Problem 17 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = 1/x $,$ x \geqslant 1 $Chris T. Numerade Educator ### Problem 18 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = 1/x $,$ 1 < x < 3 $Chris T. Numerade Educator ### Problem 19 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \sin x $,$ 0 \leqslant x < \pi /2 $Chris T. Numerade Educator ### Problem 20 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \sin x $,$ 0 < x \leqslant \pi /2 $Chris T. Numerade Educator ### Problem 21 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and tranformations of Section 1.2 and 1.3).$ f(x) = \sin x $,$ -\pi /2 \leqslant x \leqslant \pi /2 $Chris T. Numerade Educator ### Problem 22 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \cos t $,$ -3 \pi /2 \leqslant t \leqslant 3\pi /2 $Chris T. Numerade Educator ### Problem 23 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \ln x $,$ 0 < x \leqslant 2 $Chris T. Numerade Educator ### Problem 24 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = | x | $Chris T. Numerade Educator ### Problem 25 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = 1 - \sqrt{x} $Chris T. Numerade Educator ### Problem 26 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = e^x $Chris T. Numerade Educator ### Problem 27 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \left\{
\begin{array}{ll}
x^2 & \mbox{ if}-1 \leqslant x \leqslant 0\\
2 - 3x & \mbox{ if} 0 < x \leqslant 1
\end{array} \right. $Chris T. Numerade Educator ### Problem 28 Sketch the graph of$ f $by hand and use your sketch to find the absolute and local maximum and minimum values of$ f $. (Use the graphs and transformations of Section 1.2 and 1.3).$ f(x) = \left\{
\begin{array}{ll}
2x + 1 & \mbox{ if} 0 \leqslant x < 1\\
4 - 2x & \mbox{ if} 1 \leqslant x \leqslant 3
\end{array} \right. $Chris T. Numerade Educator ### Problem 29 Find the critical numbers of the function.$ f(x) = 4 + \frac{1}{3}x - \frac{1}{2}x^2 $FP Fahad P. Numerade Educator ### Problem 30 Find the critical numbers of the function.$ f(x) = x^3 + 6x^2 - 15x $Madi S. Numerade Educator ### Problem 31 Find the critical numbers of the function.$ f(x) = 2x^3 - 3x^2 - 36x $Amrita B. Numerade Educator ### Problem 32 Find the critical numbers of the function.$ f(x) = 2x^3 + x^2 + 2x $Amrita B. Numerade Educator ### Problem 33 Find the critical numbers of the function.$ g(t) = t^4 + t^3 + t^2 + 1 $Amrita B. Numerade Educator ### Problem 34 Find the critical numbers of the function.$ g(t) = | 3t - 4 | $Amrita B. Numerade Educator ### Problem 35 Find the critical numbers of the function.$ g(y) = \frac{ y - 1}{ y^2 - y + 1} $Amrita B. Numerade Educator ### Problem 36 Find the critical numbers of the function.$ h(p) = \frac{p - 1}{p^2 + 4} $Amrita B. Numerade Educator ### Problem 37 Find the critical numbers of the function$ h(t) = t^\frac{3}{4} - 2t^\frac{1}{4} $Amrita B. Numerade Educator ### Problem 38 Find the critical numbers of the function.$ g(x) = \sqrt[3]{4 - x^2} $Amrita B. Numerade Educator ### Problem 39 Find the critical numbers of the function.$ F(x) = x^\frac{4}{5} (x - 4)^2 $Amrita B. Numerade Educator ### Problem 40 Find the critical numbers of the function.$ g(\theta) = 4\theta - \tan\theta $Amrita B. Numerade Educator ### Problem 41 Find the critical numbers of the function.$ f(\theta) = 2 \cos \theta + \sin^2 \theta $Amrita B. Numerade Educator ### Problem 42 Find the critical numbers of the function.$ h(t) = 3t - \arcsin t $Amrita B. Numerade Educator ### Problem 43 Find the critical numbers of the function.$ f(x) = x^2 e^{-3x} $Amrita B. Numerade Educator ### Problem 44 Find the critical numbers of the function.$ f(x) = x^{-2} \ln x $Amrita B. Numerade Educator ### Problem 45 A formula for the derivative of a function$f$is given. How many critical numbers does$f$have?$ f'(x) = 5e^{-0.1 |x|} \sin x - 1 $Yuki H. Numerade Educator ### Problem 46 A formula for the derivative of a function$f$is given. How many critical numbers does$f$have?$ f'(x) = \frac{100 \cos^2 x}{10 + x^2} - 1 $Amrita B. Numerade Educator ### Problem 47 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = 12 + 4x - x^2 $,$ [0, 5] $Amrita B. Numerade Educator ### Problem 48 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = 5 + 54x - 2x^3 $,$ [0, 4] $Amrita B. Numerade Educator ### Problem 49 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = 2x^3 - 3x^2 - 12x + 1 $,$ [-2, 3] $Amrita B. Numerade Educator ### Problem 50 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = x^3 - 6x^2 + 5 $,$ [-3, 5] $Amrita B. Numerade Educator ### Problem 51 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = 3x^4 - 4x^3 - 12x^2 + 1 $,$ [-2, 3] $Amrita B. Numerade Educator ### Problem 52 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(t) = (t^2 - 4)^3 $,$ [-2, 3] $Amrita B. Numerade Educator ### Problem 53 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = x + \frac{1}{x} $,$ [0.2, 4] $Amrita B. Numerade Educator ### Problem 54 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = \frac{x}{x^2 - x + 1} $,$ [0, 3] $Yuki H. Numerade Educator ### Problem 55 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(t) = t - \sqrt[3]{x} $,$ [-1, 4] $Amrita B. Numerade Educator ### Problem 56 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(t) = \frac{\sqrt{t}}{1 + t^2} $,$ [0, 2] $Amrita B. Numerade Educator ### Problem 57 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(t) = 2 \cos t + \sin 2t $,$ [0, \pi /2] $Amrita B. Numerade Educator ### Problem 58 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(t) = t + \cot (t/2) $,$ [\pi /4, 7\pi /4] $Amrita B. Numerade Educator ### Problem 59 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = x^{-2} \ln x $,$ [\frac{1}{2}, 4] $Amrita B. Numerade Educator ### Problem 60 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = xe^{x/2} $,$ [-3, 1] $Amrita B. Numerade Educator ### Problem 61 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = \ln (x^2 + x + 1) $,$ [-1, 1] $Amrita B. Numerade Educator ### Problem 62 Find the absolute maximum and absolute minimum values of$f$on the given interval.$ f(x) = x - 2 \tan^{-1} x $,$ [0, 4] $Amrita B. Numerade Educator ### Problem 63 If$a$and$b$are positive numbers, find the maximum value of$ f(x) = x^a (1 - x)^b $,$ 0 \leqslant x \leqslant 1 $. Amrita B. Numerade Educator ### Problem 64 Use a graph to estimate the critical numbers of$ f(x) = | 1 + 5x - x^3 | $correct to one decimal place. Amrita B. Numerade Educator ### Problem 65 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.$ f(x) = x^5 - x^3 + 2 $,$ -1 \leqslant x \leqslant 1 $Yuki H. Numerade Educator ### Problem 66 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.$ f(x) = e^x + e^{-2x} $,$ 0 \leqslant x \leqslant 1 $Amrita B. Numerade Educator ### Problem 67 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.$ f(x) = x \sqrt{x - x^2} $Amrita B. Numerade Educator ### Problem 68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.$ f(x) = x - 2\cos x $,$ -2 \leqslant x \leqslant 0 $Amrita B. Numerade Educator ### Problem 69 After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function $$C(t) = 1.35te^{-2.802t}$$ models the average BAC, measured in mg/mL, in a group of eight male subjects$t$hours after rapid consumption of$15$ml of ethanol (corresponding to one alcoholic drink). What is the maximum average BAC during the first$3$hours? When does it occur? Source: Adapted from P. Wilkinson et. al., "Pharmacokinetics of Ethanol after Oral Administration in the Fasting State," Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207-24. Mutahar M. Numerade Educator ### Problem 70 After an antibiotic taken is taken, the concentration of the antibiotic in the bloodstream is modeled by the function $$C(t) = 8(e^{-0.4t} - e^{-0.6t})$$ where the time$t$is measured in hours and$C$is measured in$ \mu $g/mL. What is the maximum concentration of the antibiotic during the first$12$hours? Bobby B. University of North Texas ### Problem 71 Between 0$ ^\circ C $and 30$ ^\circ C $, the volume$V$(in cubic centimeters) of$1$kg of water at a temperature$T$is given approximately by the formula $$V = 999.87 - 0.06426T + 0.0085043T^3 - 0.0000679T^3$$ Find the temperature at which water has its maximum density. Mutahar M. Numerade Educator ### Problem 72 An object with weight$W$is dragged along a horizontal plane by force acting along a rope attached to the object. If the rope makes an angle$ \theta $with the plane, then the magnitude of the force is $$F = \frac{\mu W}{\mu \sin \theta + \cos \theta}$$ where$ \mu $is a positive constant called the coefficient of friction and where$ 0 \leqslant \theta \leqslant \pi /2 $. Show that$F$is minimized when$ \tan \theta = \mu $. Amrita B. Numerade Educator ### Problem 73 The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function $$L(t) = 0.01441t^3 - 0.4177t^3 + 2.703t + 1060.1$$ where$t$is measured in months since January 1, 2012. Estimate when the water level was highest during 2012. Bobby B. University of North Texas ### Problem 74 On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval$ t \in [0, 125] $. Then graph this polynomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first$125$seconds. Mutahar M. Numerade Educator ### Problem 75 When a foreign object lodged in the trachea (wind pipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction in the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the air-stream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity$v$of the air-stream is related to the radius$r$of the trachea by the equation $$v(r) = k(r_o - r) r^2$$ $$\frac{1}{2}r_o \leqslant r \leqslant r_o$$ where$k$is constant and$r_o$is the normal radius of the trachea. The restriction of$r$is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than$ \frac{1}{2}r_o $is prevented (otherwise the person would suffocate. (a) Determine the value of$r$in the interval$ [\frac{1}{2}r_o, r_o] $at which$v$has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of$v$on the interval? (c) Sketch the graph of$v$on the interval$ [0, r_o] $. Carson M. Numerade Educator ### Problem 76 Show that$5$is a critical number of the function $$g(x) = 2 + (x - 5)^3$$ but$g$does not have a local extreme value at$5$. Yuki H. Numerade Educator ### Problem 77 Prove that the function $$f(x) = x^101 + x^51 + x + 1$$ has neither a local maximum nor a local minimum. Amrita B. Numerade Educator ### Problem 78 If$f$has a local minimum value at$c$, show that the function$ g(x) = - f(x) $has a local maximum value at$c$. Amrita B. Numerade Educator ### Problem 79 Prove Fermat's Theorem for the case in which$f$has a local minimum at$c$. Amrita B. Numerade Educator ### Problem 80 A cubic function is a polynomial of degree$3$; that is, it has the form$ f(x) = ax^3 + bx^2 + cx + d$, where$ a \not= 0 \$.
(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.
(b) How many local extreme values can a cubic function have?

Amrita B.