Calculus: Early Transcendentals

Explain the difference between an absolute minimum and a local minimum.

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?

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.

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.

Use the graph to state the absolute and local maximum and minimum values of the function.

Use the graph to state the absolute and local maximum and minimum values of the function.

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 $

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 $

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 $

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.

(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 $.

(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.

(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.

(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.

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 $

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 $

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 $

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 $

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 $

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 $

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 $

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 $

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 $

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 | $

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} $

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 $

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. $

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. $

Find the critical numbers of the function.

$ f(x) = 4 + \frac{1}{3}x - \frac{1}{2}x^2 $

Find the critical numbers of the function.

$ f(x) = x^3 + 6x^2 - 15x $

Find the critical numbers of the function.

$ f(x) = 2x^3 - 3x^2 - 36x $

Find the critical numbers of the function.

$ f(x) = 2x^3 + x^2 + 2x $

Find the critical numbers of the function.

$ g(t) = t^4 + t^3 + t^2 + 1 $

Find the critical numbers of the function.

$ g(t) = | 3t - 4 | $

Find the critical numbers of the function.

$ g(y) = \frac{ y - 1}{ y^2 - y + 1} $

Find the critical numbers of the function.

$ h(p) = \frac{p - 1}{p^2 + 4} $

Find the critical numbers of the function

$ h(t) = t^\frac{3}{4} - 2t^\frac{1}{4} $

Find the critical numbers of the function.

$ g(x) = \sqrt[3]{4 - x^2} $

Find the critical numbers of the function.

$ F(x) = x^\frac{4}{5} (x - 4)^2 $

Find the critical numbers of the function.

$ g(\theta) = 4\theta - \tan\theta $

Find the critical numbers of the function.

$ f(\theta) = 2 \cos \theta + \sin^2 \theta $

Find the critical numbers of the function.

$ h(t) = 3t - \arcsin t $

Find the critical numbers of the function.

$ f(x) = x^2 e^{-3x} $

Find the critical numbers of the function.

$ f(x) = x^{-2} \ln x $

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 $

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 $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 12 + 4x - x^2 $, $ [0, 5] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 5 + 54x - 2x^3 $, $ [0, 4] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 2x^3 - 3x^2 - 12x + 1 $, $ [-2, 3] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x^3 - 6x^2 + 5 $, $ [-3, 5] $

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] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = (t^2 - 4)^3 $, $ [-2, 3] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x + \frac{1}{x} $, $ [0.2, 4] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = \frac{x}{x^2 - x + 1} $, $ [0, 3] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = t - \sqrt[3]{x} $, $ [-1, 4] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = \frac{\sqrt{t}}{1 + t^2} $, $ [0, 2] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = 2 \cos t + \sin 2t $, $ [0, \pi /2] $

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] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x^{-2} \ln x $, $ [\frac{1}{2}, 4] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = xe^{x/2} $, $ [-3, 1] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = \ln (x^2 + x + 1) $, $ [-1, 1] $

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x - 2 \tan^{-1} x $, $ [0, 4] $

If $a$ and $b$ are positive numbers, find the maximum value of $ f(x) = x^a (1 - x)^b $, $ 0 \leqslant x \leqslant 1 $.

Use a graph to estimate the critical numbers of $ f(x) = | 1 + 5x - x^3 | $ correct to one decimal place.

(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 $

(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 $

(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} $

(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 $

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.

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?

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.

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 $.

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.

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.

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] $.

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 $5$.

Prove that the function
$$ f(x) = x^101 + x^51 + x + 1 $$
has neither a local maximum nor a local minimum.

If $f$ has a local minimum value at $c$, show that the function $ g(x) = - f(x) $ has a local maximum value at $c$.

Prove Fermat's Theorem for the case in which $f$ has a local minimum at $c$.

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?