When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of 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 airstream, 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 airstream is related to the radius r of the trachea by the equation V(r) = k(r0 – r)r2 1 2 r0 ≤ r ≤ r0 where k is a constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than 1/2 r0 is prevented (otherwise the person would suffocate). (a) Determine the value of r in the interval [1/2 r0, r0] 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 f0, r0 g.
> Find the critical numbers of the function. f(x) = 2x3 + x2 + 2x
> Find the critical numbers of the function. f(x) = 2x3 – 3x2 – 36x
> Let C(t) be the concentration of a drug in the bloodstream. As the body eliminates the drug, C(t) decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus C’(t) = -kC(t), where k is a positive number called the
> Find the critical numbers of the function. f(x) = x3 + 6x2 – 15x
> 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 Sections 1.2 and 1.3.) 2x + 1 if 0 <x< 1 f(x) 4 – 2x if 1 <I< 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 Sections 1.2 and 1.3.) [x² f(x) = if -1 <x<0 if 0 <x<1 2 — Зх
> 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 Sections 1.2 and 1.3.) f (x) = ex
> 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 Sections 1.2 and 1.3.) f (x) = |x |
> Find the derivative of the function. Simplify where possible. y = tan-1(x2)
> 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 Sections 1.2 and 1.3.) f (x) = ln x, 0 , x ≤ 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 Sections 1.2 and 1.3.) f (t) = cos t, -3 π/2 ≤ t ≤ 3 π/2
> Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?
> 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 Sections 1.2 and 1.3.) f (x) = sin x, - π/2 ≤ x ≤ π/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 Sections 1.2 and 1.3.) f (x) = sin x, 0 , x ≤ π/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 Sections 1.2 and 1.3.) f (x) = sin x, 0 ≤ x , π/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 Sections 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 Sections 1.2 and 1.3.) f (x) = 1/x, x ≥ 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 Sections 1.2 and 1.3.) f (x) = 2 – 1/3x, x ≥ -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 Sections 1.2 and 1.3.) f (x) = ½ (3x – 1), x ≤ 3
> (a) Sketch the graph of a function that has two local maxima, 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.
> Find the derivative of the function. Simplify where possible. y = (tan-1x)2
> (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [21, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.
> A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after t hours. (b) Find the number of bacteria after 4 hours. (c)
> (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [21, 2] that has a local maximum but no absolute maximum.
> (a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a loc
> 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 or minimum there.
> 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 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 maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4
> Use the graph to state the absolute and local maximum and minimum values of the function. y= g(x) -1 1
> Use the graph to state the absolute and local maximum and minimum values of the function. yA y = f(x) 1 1
> Find an equation of the tangent line to the hyperbola x2 / a2 - y2 / b2 = 1 at the point (x0, y0).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. An equation of the tangent line to the parabola y = x2 at (-2, 4) is y - 4 = 2x(x + 2).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a rational function is a rational function.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. . If f (x) = (x6 – x4)5, then f (31)(x) = 0.
> The cost, in dollars, of producing x units of a certain commodity is C(x) = 920 + 2x - 0.02x2 + 0.00007x3 (a) Find the marginal cost function. (b) Find C’(100) and explain its meaning. (c) Compare C’(100) with the cost of producing the 101st item.
> Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a polynomial is a polynomial.
> Show that the length of the portion of any tangent line to the astroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is constant.
> The Michaelis-Menten equation for the enzyme chymotrypsin is v = 0.14[S] / 0.015 + [S] where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dv/d[S] and interpret it.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If y = e2, then y’ = 2e.
> Suppose f is a differentiable function such that f (g(x)) = x and f ‘(x) = 1 + [f(x)]2 . Show that g’(x) = 1 / (1 + x2).
> Express the limit as a derivative and evaluate. lim ℎ→0 4 16+ℎ −2 ℎ
> A window has the shape of a square surmounted by a semi- circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of th
> Evaluate dy if y = x3 - 2x2 + 1, x = 2, and dx = 0.2.
> The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is π/6?
> Prove that the function f (x) = x101 + x51 + x + 1 has neither a local maximum nor a local minimum.
> A waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp? 4 ft -15 ft–
> Show that 5 is a critical number of the function t(x) = 2 + (x – 5)3 but t does not have a local extreme value at 5.
> Find the derivative of the function. Simplify where possible. H(t) = cot-1(t) + cot-1(1/t)
> 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 je
> 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) = 0.01441t3 - 0.4177t2 + 2.703t + 1060.1 where t is measured in months since January 1, 2012. Estimate when the water
> Between 0°C and 30°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 + 0.0085043T2 - 0.0000679T3 Find the temperature at which water has its maximum density.
> After an antibiotic tablet 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 mg/mL. What is the maximum concentration of the an
> 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) = 1.35
> (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 ≤ x ≤ 0
> A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?
> (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) = ex + e-2x, 0 ≤ x ≤ 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) = x5 - x3 + 2, -1 ≤ x ≤ 1
> Find the derivative of the function. Simplify where possible. y = tan-1 (x –( 1 + x2 )
> Use a graph to estimate the critical numbers of f (x) = |1 + 5x - x3 | correct to one decimal place.
> If a and b are positive numbers, find the maximum value of f (x) = xa(1 – x)b, 0 ≤ x ≤ 1.
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = x - 2 tan-1x, [0, 4]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = ln(x2 + x + 1), [-1, 1]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = xex/2, [-3, 1]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = x-2 ln x , [1/2, 4]
> A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3/s, how fast is the water level rising when the water is 5 cm deep?
> Find the absolute maximum and absolute minimum values of f on the given interval. f (t) = t + cot (t/2), [π/4, 7π/4]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (t) = 2cos t + sin 2t, [0, π/2]
> Find the derivative of the function. Simplify where possible. F(x) = x sec-1(x3)
> Find the absolute maximum and absolute minimum values of f on the given interval. f (t) = (t2 - 4)3, [-2, 3]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = 3x4 - 4x3 - 12x2 + 1, [-2, 3]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = x3 - 6x2 + 5, [-3, 5]
> The volume of a cube is increasing at a rate of 10 cm3/min. How fast is the surface area increasing when the length of an edge is 30 cm?
> If f has a local minimum value at c, show that the function g(x) = -f (x) has a local maximum value at c.
> Find equations of the tangent line and normal line to the curve at the given point. x2 + 4xy + y2 = 13, (2, 1)
> Find an equation of the tangent to the curve at the given point. y = 4 sin2x, ( π/6, 1)
> Use mathematical induction (page 72) to show that if f (x) = xex, then f(n)(x) = (x + n)ex. Mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle. Principle of Mathem
> Find f(n)(x) if f (x) = 1/(2 – x).
> Find y’’ if x6 + y6 = 1.
> Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a) Graph the curve with equation y( y2 – 1)(y – 2)d = x(x – 1)(x – 2) At how many points does this curve have horizontal tangents? Estimate the x-co
> If g(θ) = θ sin θ, find g’’( π/6).
> Find all values of c such that the parabolas y = 4x2 and x = c + 2y2 intersect each other at right angles.
> Calculate y’. y = cosh-1 (sinh x)
> Calculate y’. y = ln (cosh 3x)
> Calculate y’. y = (sin mx) / x
> Calculate y’. y = x sinh (x2)
> If x2 + xy + y3 = 1, find the value of y’’’ at the point where x − 1.
> The figure shows a circle with radius 1 inscribed in the parabola y = x2. Find the center of the circle. y 4 y=x?
> Calculate y’. xey = y - 1
> Calculate y’. y = tan2 (sin θ)
> Calculate y’. y = cot (3x2 + 5)
> Calculate y’. y = 10tan πθ
> If xy + ey = e, find the value of y’’ at the point where x = 0.
> Calculate y’. y = ecos x + cos (ex)
> Calculate y’. y = x tan-1 (4x)
> Calculate y’. y = (cox x)x
> Calculate y’. sin (xy) = x2 - y
> Calculate y’. y = (1 – x-1)-1
> Find y’’ by implicit differentiation. x3 - y3 = 7
