(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 rate of 1m/s, how fast is the area of the spill increasing when the radius is 30 m?
> (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?
> 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?
> 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 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
> 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
> Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = In(x² + x + 1), [-1, 1]
> Find the absolute maximum and absolute minimum values of f on the given interval. fa) — 2x* — 3x? — 12х + 1, [-2, 3]
> Find the critical numbers of the function. g (x) = x1/3 – x-2/3
> A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function where is measured in years since August of 1993. Estimate the times when sugar was cheapest and most expensive during the period 1993–
> An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is Where µ is a positive constant
> (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) = (ln x)/√x
> f (x) = ex3-x, -1 < 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.
> f (x) = x5 - x3 + 2, -1 < 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.
> Use a graph to estimate the critical numbers of f (x) = |x3 – 3x2 + 2| correct to one decimal place.
> (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) = cos2 x – 2 sin x, 0 < x < 2π
> If and are positive numbers, find the maximum value of f (x) = xa (1 – x) b, 0 < x < 1.
> Suppose f (3) = 2, f'(3) = ½ and f'(x) > 0 and f"(x) < 0 for all x. (a). Sketch a possible graph for f. (b). How many solutions does the equation f (x) = 0 have? Why? (c). Is it possible that f'(2) = 1/3? Why?
> Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = t + cot (t/2), [7/4, 77/4]
> Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = x – In x, 1, 2]
> Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = xe, [-1,4] -1/8
> A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
> Find the critical numbers of the function. f (x) = x3 + 3x2 - 24x
> Find the absolute maximum and absolute minimum values off on the given interval. f(x) = (x² – 1)', [-1, 2] %3D
> Find the absolute maximum and absolute minimum values of f on the given interval.
> The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock?
> (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) = x2 lnx
> At noon, ship A is 150 km west of ship B. Ship A is sailing east 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). What quantities are given in the problem? (b). What is the unknown? (
> (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) = sin x + cos x, 0 < x < 2π
> (a). Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b). Estimate the value of x at which f increases most rapidly. Then find the exact value.
> (a). Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b). Estimate the value of x at which f increases most rapidly. Then find the exact value.
> Use the methods of this section to sketch the curve y = x3 – 3a2x + 2a3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
> Find the absolute maximum and absolute minimum values off on the given interval.
> A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking?
> A formula for the derivative of a function f is given. How many critical numbers does f have?
> (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.
> Explain the difference between an absolute minimum and a local minimum.
> The graph of the derivative f' of a function f is shown. (a). On what intervals is f increasing? Decreasing? (b). At what values of x does f have a local maximum? Local minimum? (c). If it is known that f (0) = 0, sketch a possible graph of
> Let f (x) = x3 - x. In Examples 3 and 7 in Section 2.7, we showed that f'(x) = 3x2 - 1 and f"(x) = 6x. Use these facts to find the following. (a). The intervals on which f is increasing or decreasing. (b). The intervals on which f is concave upward or do
> Where does the normal line to the parabola y = x – x2 at the point (1, 0) intersect the parabola a second time? Illustrate with a sketch.
> Where is the greatest integer function f (x) = [[x]] not differentiable? Find a formula for f' and sketch its graph.
> (a). Use a graphing calculator or computer to graph the function f (x) = x4 – 3x2 – 6x2 + 7x + 30 in the viewing rectangle [-3, 5] by [-10, 50]. (b). Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f'. (See E
> Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely?
> The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle θ. Find lim 0t d d
> The function f (x) = sin (x + sin 2x), 0 < x < π, arises in applications to frequency modulation (FM) synthesis. (a). Use a graph of f produced by a graphing device to make a rough sketch of the graph of f'. (b). Calculate f (x) and use this expression,
> Find constants A and B such that the function y = A sin x + B cos x satisfies the differential equation y" + y' – 2y = sin y.
> Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4, 3).
> Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it.
> Graph the function f (x) = x + √|x|. Zoom in repeatedly, first toward the point (-1, 0) and then toward the origin. What is different about the behavior of f in the vicinity of these two points? What do you conclude about the differentiability of f?
> An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is where µ is a constant called the
> Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a). Use a CAS to find the derivative in Example 5 and compare with the
> The cycloid was x = r (θ – sin θ), y = r (1 – cos θ) discussed in Example 7 in Section 1.7. (a). Find an equation of the tangent to the cycloid at the point where θ = π/3. (b). At what points is the tangent horizontal? Where is it vertical? (c). Graph th
> (a). If f (x) = (x2 – 1)/ (x2 + 1), find f'(x) and f"(x). (b). Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f', and f".
