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 the cycloid and its tangent lines for the case r = 1.
> 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
> (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".
> Show that the curve with parametric equations x = sin t, y = sin (t + sin t) has two tangent lines at the origin and find their equations. Illustrate by graphing the curve and its tangents.
> An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t + 3 sin t, t > 0, where s is measured in centimeters and
> Let P (t) be the percentage of Americans under the age of 18 at time t. The table gives values of this function in census years from 1950 to 2000. (a). What is the meaning of P'(t)? What are its units? (b). Construct a table of estimated values for P'(
> 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 f. у. y
> Let be the tangent line to the parabola y = x2 at the point (1, 1). The angle of inclination of l is the angle ø that l makes with the positive direction of the x-axis. Calculate ø correct to the nearest degree.
> 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.
> (a). The graph of a position function of a car is shown, where s is measured in feet and t in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at t = 10 seconds? (b). Use the acceleration curve from part (a) t
> A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0
> If the equation of motion of a particle is given by s = A cos (wt + δ), the particle is said to undergo simple harmonic motion. (a). Find the velocity of the particle at time t. (b). When is the velocity 0?
> The table gives the US population from 1790 to 1860. (a). Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit? (b). Estimate the rates of population grow
> The displacement of a particle on a vibrating string is given by the equation s (t) = 10 + 1/4 sin (10πt) where is measured in centimeters and in seconds. Find the velocity of the particle after seconds.
> Find the 1000th derivative of f (x) = xe-x.
> Find the 50th derivative of y = cos 2x.
> For what values of r does the function y = erx satisfy the differential equation y" – 4y' + y = 0?
> Show that the function y = e2x (A cos 3x + B sin 3x) satisfies the differential equation y" – 4y' + 13y = 0.
> If F (x) = f (xf (xf (x))), where f (1) = 2, f (2) = 3, f'(1) = 4, f'(2) = 5, and f'(3) = 6, find F'(1).
> If F (x) = f (3f (4f (x))), where f (0) = 0 and f'(0) = 2, find F'(0).
> If g is a twice differentiable function and f (x) = xg (x2), find f" in terms of g, g', and g".
> Let r (x) = f (g (h (x))), where h (1) = 2, g (2) = 3, h'(1) = 4, g'(2) = 5, and f'(3) = 6. Find r'(1).
> Suppose f is differentiable on R and a is a real number. Let F (x) = f (xa) and G (x) = [f (x)]a. Find expressions for (a) F'(x) and (b) G'(x).
> A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q = f (p). Then the total revenue earned with selling
> Suppose f is differentiable on R. Let F (x) = f (ex) and G (x) = ef(x). Find expressions for (a) F'(x) and (b) G'(x).
> If g (x) = f (f (x)), use the table to estimate the value of g'(1). 0.0 0.5 1.0 1.5 2.0 2.5 f(x) 1.7 1.8 2.0 2.4 3.1 4.4
> Use the table to estimate the value of h'(0,5), where h (x) = f (g (x)). 0.1 0.2 0.3 0.4 0.5 0.6 f(x) 12.6 14.8 18.4 23.0 25.9 27.5 29.1 g(x) 0.58 0.40 0.37 0.26 0.17 0.10 0.05
> If f is the function whose graph is shown, let h (x) = f (f (x)) and g (x) = f (x2). Use the graph of f to estimate the value of each derivative. (a). h'(2) (b). g'(2) y= f(x) 1 1
> The Bessel function of order 0, y = j (x), satisfies the differential equation xy" + y' + xy = 0 for all values of and its value at 0 is j (0) = 1. (a). Find j'(0). (b). Use implicit differentiation to find j''(0).
> Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3).
> Find all points on the curve x2y2 + xy = 2 where the slope of the tangent line is -1.
> If h (x) = √4 + 3f (x), where f (1) = 7 and f'(1) = 4, find h'(1).
> If F (x) = f (g (x)), where f (-2)- 8, f'(-2) = 4, f'(5) = 3, g (5) = -2, and g'(5) = 6, find F'(5).
> Show that the sum of the - and -intercepts of any tangent line to the curve √x + √y = √c is equal to c.
> In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per yea
> Find all points on the graph of the function f (x) = 2 sin x sin2x at which the tangent line is horizontal.
> Show that limx→∞ (1 + x/n) n for any x > 0.
> Use the definition of derivative to prove that limx→0 ln (1 + x)/x = 1
> Find d9/dx9 (x8 ln x).
> Find a formula for f(n)(x) if f (x) = ln (x - 1).
> Find y' if xy = yx.
> Find y' if y = ln (x2 + y2).
> Use logarithmic differentiation to find the derivative of the function. y = (sin x) lnx
> Use logarithmic differentiation to find the derivative of the function. y = (tan x)1/x
> Use logarithmic differentiation to find the derivative of the function. y = √xx
> Use the definition of a derivative to find f'(x) and f''(x). Then graph f, f', and f'' on a common screen and check to see if your answers are reasonable. f (x) = x3 - 3x
> Use logarithmic differentiation to find the derivative of the function. y = (cos x) x
> Use logarithmic differentiation to find the derivative of the function. y = x cos x
> Use logarithmic differentiation to find the derivative of the function. y = xx
> Use logarithmic differentiation to find the derivative of the function. y = 4√ (x2 + 1/x2 – 1)
> Use logarithmic differentiation to find the derivative of the function. y = sin2x tan4x/ (x2 + 1)2
> Use logarithmic differentiation to find the derivative of the function. y = √x ex2 (x2 + 1)10
> When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: F = kR4 (This is known as Poiseuille’s Law; we will show why it i
