(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 opposite points on the equator that have exactly the same temperature. (b). Does the result in part (a) hold for points lying on any circle on the earth’s surface? (c). Does the result in part (a) hold for barometric pressure and for altitude above sea level?
> 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?
> (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)?
> 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 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?
> Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. f(x) = Vx4 – 2x2 + 2 – 2, -3 <I<3
> Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. sin x f(x) = 1+ x?"
> The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.
> The graph of a function f is shown. Which graph is an antiderivative of f and why? a
> Let f (x) = x4 – 2x2. (a). Use the definition of a derivative to find f'(x) and f"(x). (b). On what intervals is f increasing or decreasing? (c). On what intervals is f concave upward or concave downward?
> If f'(x) = e-x2, what can you say about f?
> For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?
> Sketch the graph of a function that satisfies all of the given conditions. f'(x) > 0 if |x| < 2, f'(x) < 0 if |x| > 2, f'(-2) = 0, lim |f'(x) | = ∞, f"(x) > 0 if x # 2
> Sketch the graph of a function that satisfies all of the given conditions. f'(1) = f'(-1) = 0, f'(x) < 0 if |x| < 1, f'(x) > 0 if 1 < |x|< 2, f'(x) = -1 if |x|> 2, f"(x) < 0 if –2 <x< 0, inflection point (0, 1)
> Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(2) = f'(4) = 0, f'(x) > 0 if x < 0 or 2 <x< 4, f'(x) < 0 if 0 <x< 2 or x> 4, f"(x) > 0 if 1 < x< 3, f"(x) < 0 if x < 1 or x > 3
> Sketch the graph of a function that satisfies all of the given conditions. f'(x) > 0 for all x # 1, vertical asymptote x = 1, f"(x) > 0 if x < 1 or x> 3, f"(x) < 0 if 1 <x<3
> Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(4) = 0, f'(x) > 0 if x < 0, f'(x) < 0 if 0 < x < 4 or if x > 4, f"(x) > 0 if 2 < x< 4, f"(x) < O if x < 2 or x>4
> Sketch the graph of a function whose first derivative is always negative and whose second derivative is always positive.
> Sketch the graph of a function whose first and second derivatives are always negative.
> The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its derivatives.
> (a). Sketch a curve whose slope is always positive and increasing. (b). Sketch a curve whose slope is always positive and decreasing. (c). Give equations for curves with these properties.
> Let P (x1, y1) be a point on the parabola y2 = 4px with focus F (p, 0). Let a be the angle between the parabola and the line segment FP, and let β be the angle between the horizontal line y = y' and the parabola as in the figure. Prove that a
> Estimate the value of f'(a) by zooming in on the graph off. Then differentiate f to find the exact value of f'(a) and compare with your estimate. f (x) = 1/√x, a = 4
> If limx→a [f (x) + g (x)] = 2 and limx→a [f (x) - g (x)] = 1, find limx→a [f (x) g (x)].
> If f is a differentiable function and g (x) = x f (x), use the definition of a derivative to show that g'(x) = x f'(x) + f (x).
> For what value of does the equation e2x = k√x have exactly one solution?
> If y = ((x/√a2 – 1) – (2/√a2 – 1) arctan sin x/a + (√a2 – 1) + cos x) show that y' = 1/a + cos x.
> If f and g are differentiable functions with f (0) = g (0) = 0 and g'(0) ≠0, show that f(x) _ f'(0) lim g(x) g'(0)
> If [[x]] denotes the greatest integer function, find limx→∞ x/[[x]].
> Find the values of the constants a and b such that limx→0 3√ax + b – 2/x = 5/12
> Show that dn/dxn = eax sin bx) = rneaxs sin (bx + nθ) where a and b are positive numbers, r2 = a2 + b2, and θ = tan-1(b/a).
> Tangent lines T1 and T2 are drawn at two points P1 and P2 on the parabola y = x2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2; it intersects T1 at Q1 and T2 at Q2. Show that | PQ.| | P | PQ:| 1
> Find all values of a such that f is continuous on R: Jx +1 if x< a f(x) = |x? - if x>a
> Find a cubic function y = ax2 + bx2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2, 6).
> If f is differentiable at a, where a > 0, evaluate the following limit in terms of f'(a): f(x) – f(a) lim VI - Va x,
> If f (x) = limt→x sec t – sec x/t - x, find the value of f'(π/4).
> Show that d/dx (sin2 x/ 1 + cot x + cos2x/1 + tan x) = -cos 2x
> Find numbers a and b such that limx→1 √ax + b – 2/x = 1
> Evaluate limx→1 3√x – 1/√x - 1
> Bézier curves are used in computer-aided design and are named after the French mathematician Pierre Bézier (1910–1999), who worked in the automotive industry. A cubic Bézier curve is determined by
> The tangent line approximation L (x) is the best first-degree (linear) approximation f (x) to near x = a because f (x) and L (x) have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a s
> Certain combinations of the exponential functions, ex and e-x arise so frequently in mathematics and its applications that they deserve to be given special names. This project explores the properties of functions called hyperbolic functions. The hyperbol
> State each differentiation rule both in symbols and in words. (a). The Power Rule (b). The Constant Multiple Rule (c). The Sum Rule (d). The Difference Rule (e). The Product Rule (f). The Quotient Rule (g). The Chain Rule
> (a). Suppose the horizontal distance between P and Q is 100 ft. Write equations in a, b, and that will ensure that the track is smooth at the transition points. (b). Solve the equations in part (a) for a, b, and c to find a formula for f (x). (c). Plot L
> Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 – 3x. Find the values of a, b, c, and d.
> The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L1 (x) for x 100] doesn’t have a continuous second derivative. So you decide to improve the design by using
