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?
> A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?
> A (x) = x√x + 3 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check your
> Each side of a square is increasing at a rate of 6m/s. At what rate is the area of the square increasing when the area of the square is 16 cm2?
> 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.) (4 — х? if -2 <x<0 |2х — 1 if 0 <x <2 if -2 <х<0 f(x) =
> Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = t/4 – 1², [-1, 2]
> Use the graph to state the absolute and local maximum and minimum values of the function. |y= f(x) 1t 1
> Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10
> A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 ft3/min, how fast is the water level rising when the depth a
> A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft3 /min, how fast is the water level rising when the water is 6
> Water is leaking out of an inverted conical tank at a rate of 10,000 cm3 /min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate
> How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?
> For what values of c is the function increasing on (-∞, ∞)? f (x) = cx + 1/x2 + 3
> (a). If the function f (x) = x3 + ax2 + bx has the local minimum value -2/9√3 at 1/√3, what are the values of a and b? (b). Which of the tangent lines to the curve in part (a) has the smallest slope?
> For what values of does the polynomial P (x) = x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as decreases?
> Find the critical numbers of the function. f (x) = x3 + x2 + x
> (a). Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b). Use a graph of f" to give better estimates. f (x) = x3(x – 2)4
> 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
> (a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = 2x3 + 3x2 - 36x
> Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2 and x3, show that the x-coordinate of the inflection point is (x1 + x2 + x3)/3.
> Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain t
> Show that the curves y = e-x and y = -e-x touch the curve y = e-x sin x at its inflection points.
> The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2?
> At 2:00 PM a car’s speedometer reads 30 mi/h. At 2:10 PM it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h2.
> Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f (t) = g (t) – h (t), where and are the position functions of the two runners.]
> Suppose that 3 < f'(x) < 5 for all values of x. Show that 18 < f (8) – f (2) < 30.
> Suppose that f (0) = -3 and f'(x) < 5 for all values of x. The inequality gives a restriction on the rate of growth of f, which then imposes a restriction on the possible values of f. Use the Mean Value Theorem to determine how large f (4) can possibly b
> f (x) = 2 + 2x2 - x4 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check
> (a). Show that ex > 1 + x for x > 0. (b). Deduce that ex > 1 + x + 1/2x2 for x > 0. (c). Use mathematical induction to prove that for x > 0 and any positive integer n, e>1+x + +- n! + 2!
> f (x) = ln (x4 + 27) (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check
> h (x) = (x + 1)5 - 5x - 2 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. C
> Between 00C and 300C, 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.
> f (x) = x - 2 cos x, -2 < x < 0 (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.
> The graph of the first derivative f' of a function f is shown. (a). On what intervals is f increasing? Explain. (b). At what values of x does f have a local maximum or minimum? Explain. (c). On what intervals is f concave upward or concave downward? Ex
> A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. (a). At what rate is his distance from second base decreasing when he is halfway to first base? (b). At what rate is his distance
> f (x) = x√x - x2 (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.
> The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant µ is called the mean and the positive constant σ is called the standard deviation. For simplicity,
> A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S (t) = Atp e-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual de
> (a). Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b). Use a graph of f" to give better estimates. f(x) = cos x + cos 2x, 0 <x< 27
> Coulomb’s Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles wit
> The figure shows a beam of length L embedded in concrete walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve where E and I are positive constants. (E is Young’s modulus
> 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. y. 0 a b c drs x
> A formula for the derivative of a function f is given. How many critical numbers does f have? f'(x) = 5e-01|=| sinx – 1
> Find the critical numbers of the function. f (x) = x-2 ln x
> Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> f (x) = 2 + 3x - x3 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check y
> f (x) = 2x3 - 3x2 - 12x (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Che
> 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, -37/2 <t< 37/2
> (a). Find the critical numbers of f (x) = x4(x – 1)3. (b). What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c). What does the First Derivative Test tell you?
> Find the critical numbers of the function. h (t) = 3t - arcsin t
> 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
> Suppose f" is continuous on (-∞, ∞). (a). If f'(2) = 0 and f"(2) = -5, what can you say about f? (b). If f'(6) = 0 and f"(6) = 0, what can you say about f?
> Show that the curve y = (1 + x)/ (1 + x2) has three points of inflection and they all lie on one straight line.
> Show that tan x > x for 0 < x < π/2. [ Show that f (x) = tan x - x is increasing on (0, π/2).]
> For what values of the numbers and does the function have the maximum value f (2) = 1? f(x) = axe" ахе
> Find a cubic function f (x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at = -2 and a local minimum value of 0 at x= 1.
> Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer? f(x) = x? + 4 -2
> In the theory of relativity, the energy of a particle is where m0 is the rest mass of the particle, is its wave length, and h is Planck’s constant. Sketch the graph of E as a function of λ. What does the graph say about the
> A particle moves along the curve y = √1 + x2. As it reaches the point (2, 3), the y-coordinate is increasing at a rate of 4cm/s. How fast is the x-coordinate of the point changing at that instant?
> In the theory of relativity, the mass of a particle is where m0 is the rest mass of the particle m, is the mass when the particle moves with speed relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v. mo m
> Find dy/dx and d2y/dx2. For which values of is the parametric curve concave upward? x = cos 2t, y = cos t, 0 <t< T
> Find dy/dx and d2y/dx2. For which values of is the parametric curve concave upward? x=t - 12t, y = t² – 1 = 1? – 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) = x², 0<x< 2
> Find the critical numbers of the function. f (θ) = 2 cos θ + sin2 θ
> Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f". tan-x f(x) 1+ x' .3
> Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f". x* + x' + 1 f(x) Vx? + x + 1
> Suppose the derivative of a function f is f'(x) = (x + 1)2 (x – 3)5 (x – 6)4. On what interval is f increasing?
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> g (x) = 200 + 8x3 + x4 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Chec
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> If V is the volume of a cube with edge length and the cube expands as time passes, find dV/dt in terms of dx/dt.
> Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer? f(x) = x + VI - 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 Sections 1.2 and 1.3.) f(x) = 2 – x, x> -2
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> Find the critical numbers of the function. g (θ) = 4 θ - tan θ
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check you
> Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 2cos t + sin 2t, [0, 7/2]
> (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check you
> h (x) = x5 - 2x3 + x (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check
> (a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = e2x + e-x
> (a). State the First Derivative Test. (b). State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?
> Suppose you are given a formula for a function f. (a). How do you determine where f is increasing or decreasing? (b). How do you determine where the graph of f is concave upward or concave downward? (c). How do you locate inflection points?
> Use the given graph of to find the following. (a). The open intervals on which f is concave upward (b). The open intervals on which f is concave downward (c). The coordinates of the points of inflection F1-
> 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) = }(3x – 1), I< 3 %3D
> Find the absolute maximum and absolute minimum values of f on the given interval. fx) — х* — 6х? + 9х + 2, [-1,4]
> Use the graph of f to estimate the values of that satisfy the conclusion of the Mean Value Theorem for the interval [0, 8]. y = f(x)
> (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci
> Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = x – 2 tan'x, [0, 4]
> (a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = √xe-x
> The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?
> A particle is moving along the curve y = √x. As the particle passes through the point (4, 2), its x-coordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin changing at this instant?
> At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?
> A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is
> (a). If A is the area of a circle with radius and the circle expands as time passes, find dA/dt in terms of dr/dt. (b). Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant ra
> A cubic function is a polynomial of degree 3; that is, it has the form f (x) = ax3 + bx2 + cx + d, where a ≠ 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.
> A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? (a). What quantities are given in t
> 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 ai
