(a). Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness ∆r. (b). What is the error involved in using the formula from part (a)?
> 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
> One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 300, with a possible error of ±10. (a) Use differentials to estimate the error in computing the length of the hypotenuse. (b) What is the percentage error?
> Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.
> Sketch the parabolas y = x2 and y = x2 – 2x + 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
> The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a). Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b). Use differentials to estimate the maximum error
> The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a). Use differentials to estimate the maximum error in the calculated area of the disk. (b). What is the relative error? What is the percentage error?
> Find an equation of the tangent line to the curve at the given point. y = ln (xex2), (1, 1)
> Let y = √x. (a). Find the differential dy. (b). Evaluate dy and ∆y if x= 1 and dx = ∆x = 1. (c). Sketch a diagram like Figure 6 showing the line segments with lengths dx, dy, and ∆y.
> Let y = ex/10. (a). Find the differential dy. (b). Evaluate dy and ∆y if x = 0 and dx = 0.1.
> Find the differential of each function. (a). y = etan πt (b). y = √1 + ln z
> Find the differential of each function. (a). u + 1/u - 1 (b). y = (1 + r3)-2
> If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value [A]= [B] = a moles/L, then [C] a2kt/ (akt + 1) where k is a co
> Explain, in terms of linear approximations or differentials, why the approximation is reasonable. ln 1.05 ≈ 0.05
> Explain, in terms of linear approximations or differentials, why the approximation is reasonable. (1.01)6 ≈ 1.06
> If c > 1/2, how many lines through the point (0, c) are normal lines to the parabola y = x2? What if c < 1/2?
> Explain, in terms of linear approximations or differentials, why the approximation is reasonable. sec 0.08 ≈ 1
> If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after minutes as. Find the rate at which water is draining from th
> Use a linear approximation (or differentials) to estimate the given number. (8.06)2/3
> Differentiate the function. H (z) = ln √a2 – z2/a2 + z2
> Use a linear approximation (or differentials) to estimate the given number. (2.001)5
> Verify the given linear approximation at a = 0. Then determine the values of for which the linear approximation is accurate to within 0.1. ex ≈ 1 + x
> Verify the given linear approximation at a = 0. Then determine the values of for which the linear approximation is accurate to within 0.1. 1/ (1 + 2x)4 ≈ 1 - 8x
> Verify the given linear approximation at a = 0. Then determine the values of for which the linear approximation is accurate to within 0.1. tan x ≈ x
> Verify the given linear approximation at a = 0. Then determine the values of for which the linear approximation is accurate to within 0.1. 式
> Find the linear approximation of the function g (x) = 3√1 + x at a = 0 and use it to approximate the numbers 3√0.95 and 3√1.1. Illustrate by graphing g and the tangent line.
> Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y = x2. Where do these lines intersect?
> Find the linear approximation of the function f (x) = √1 - x at a = 0 and use it to approximate the numbers √0.9 and √0.99. Illustrate by graphing f and the tangent line.
> Find the linearization L (x) of the function at a. f (x) = x3/4, a = 16
> Find the linearization L (x) of the function at a. f (x) = cos x, a = π/2
> Find the linearization L (x) of the function at a. f (x) = ln x a = 1
> Find the linearization L (x) of the function at a. f (x) = x4 + 3x2, a = -1
> The table shows the population of Nepal (in millions) as of June 30 of the given year. Use a linear approximation to estimate the population at midyear in 1989. Use another linear approximation to predict the population in 2010. 1985 1990 1995 2000
> Differentiate the function. f (x) = sin (ln x)
> Atmospheric pressure P decreases as altitude h increases. At a temperature of 150C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h = 1 km, and 74.9 kPa at h = 2 km. Use a linear approximation to estimate the atmospheric pressure at a
> Explain why the natural logarithmic function y = ln x is used much more frequently in calculus than the other logarithmic functions y = logax.
> Find points P and Q on the parabola y = 1 -x2 so that the triangle ABC formed by the x-axis and the tangent lines at P and Q is an equilateral triangle. (See the figure.) y A P. B
> The figure shows the graphs of f, f', and f''. Identify each curve, and explain your choices. у. a b
> For which positive numbers is it true that ax > 1 + x for all x?
> Suppose f is a function that satisfies the equation f (x + y) = f (x) + f (y) + x2y + xy2 for all real numbers x and y. Suppose also that limx→0 f (x)/x = 1 (a). Find a (0). (b). Find f'(0). (c). Find f'(x).
> Water is flowing at a constant rate into a spherical tank. Let V (t) be the volume of water in the tank and H (t) be the height of the water in the tank at time t. (a). What are the meanings of V'(t) and H'(t)? Are these derivatives positive, negative, o
> Let T and N be the tangent and normal lines to the ellipse x2/9 + y2/4= 1 at any point P on the ellipse in the first quadrant. Let xT and yr be the x- and y-intercepts of T and xN and yN be the intercepts of N. As P moves along the ellipse in the first q
> (a). The figure shows an isosceles triangle ABC with ∠B = ∠C. The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fixed but the altitude |AM| of the triangle approaches 0, so
> (a). If we start from 00 latitude and proceed in a westerly direction, we can let T (x) denote the temperature at the point at any given time. Assuming that T is a continuous function of x, show that at any fixed time there are at least two diametrically
> A fixed point of a function f is a number c in its domain such that f (c) = c. (The function doesn’t move c; it stays fixed.) (a). Sketch the graph of a continuous function with domain [0, 1] whose range also lies in [0, 1]. Locate a fixed point of f. (b
> The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the -axis as the wheel rotates counter clockwise at a rate of 360 revolutions per minute. (a). Find the angular velocit
> Sketch the region in the plane defined by each of the following equations. (a) [x]² + [y]² = 1 (b) [x]² – []° = 3 (c) [r + y]² = 1 (d) [x] + [y] = 1
> Evaluate limx→0 |2x -1|- |2x + 1|/x
> A tangent line is drawn to the hyperbola xy = c at a point P. (a). Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b). Show that the triangle formed by the tangent line and the coordinate axes always ha
> The graphs of a function f and its derivative f' are shown. Which is bigger, f' (-1) or f"(1)? у.
> The figure shows a point P on the parabola and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it. y y=x P
> Show that the tangent lines to the parabola y = ax2 + bx + c at any two points with x-coordinates p and q must intersect at a point whose -coordinate is halfway between p and q.
> Find the point where the curves y = x3 – 3x + 4 and y = 3 (x2 – x) are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.
> The figure shows a circle with radius 1 inscribed in the parabola y = x2. Find the center of the circle. y y= x?
