Find y’’ by implicit differentiation. sin y + cos 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. f (x) = x5 - x3 + 2, -1 ≤ x ≤ 1
> Find the derivative of the function. Simplify where possible. y = tan-1 (x –( 1 + x2 )
> Use a graph to estimate the critical numbers of f (x) = |1 + 5x - x3 | correct to one decimal place.
> If a and b are positive numbers, find the maximum value of f (x) = xa(1 – x)b, 0 ≤ x ≤ 1.
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = x - 2 tan-1x, [0, 4]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = ln(x2 + x + 1), [-1, 1]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = xex/2, [-3, 1]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = x-2 ln x , [1/2, 4]
> A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3/s, how fast is the water level rising when the water is 5 cm deep?
> Find the absolute maximum and absolute minimum values of f on the given interval. f (t) = t + cot (t/2), [π/4, 7π/4]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (t) = 2cos t + sin 2t, [0, π/2]
> Find the derivative of the function. Simplify where possible. F(x) = x sec-1(x3)
> Find the absolute maximum and absolute minimum values of f on the given interval. f (t) = (t2 - 4)3, [-2, 3]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = 3x4 - 4x3 - 12x2 + 1, [-2, 3]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = x3 - 6x2 + 5, [-3, 5]
> The volume of a cube is increasing at a rate of 10 cm3/min. How fast is the surface area increasing when the length of an edge is 30 cm?
> If f has a local minimum value at c, show that the function g(x) = -f (x) has a local maximum value at c.
> Find equations of the tangent line and normal line to the curve at the given point. x2 + 4xy + y2 = 13, (2, 1)
> Find an equation of the tangent to the curve at the given point. y = 4 sin2x, ( π/6, 1)
> Use mathematical induction (page 72) to show that if f (x) = xex, then f(n)(x) = (x + n)ex. Mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle. Principle of Mathem
> Find f(n)(x) if f (x) = 1/(2 – x).
> Find y’’ if x6 + y6 = 1.
> Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a) Graph the curve with equation y( y2 – 1)(y – 2)d = x(x – 1)(x – 2) At how many points does this curve have horizontal tangents? Estimate the x-co
> If g(θ) = θ sin θ, find g’’( π/6).
> Find all values of c such that the parabolas y = 4x2 and x = c + 2y2 intersect each other at right angles.
> Calculate y’. y = cosh-1 (sinh x)
> Calculate y’. y = ln (cosh 3x)
> Calculate y’. y = (sin mx) / x
> Calculate y’. y = x sinh (x2)
> If x2 + xy + y3 = 1, find the value of y’’’ at the point where x − 1.
> The figure shows a circle with radius 1 inscribed in the parabola y = x2. Find the center of the circle. y 4 y=x?
> Calculate y’. xey = y - 1
> Calculate y’. y = tan2 (sin θ)
> Calculate y’. y = cot (3x2 + 5)
> Calculate y’. y = 10tan πθ
> If xy + ey = e, find the value of y’’ at the point where x = 0.
> Calculate y’. y = ecos x + cos (ex)
> Calculate y’. y = x tan-1 (4x)
> Calculate y’. y = (cox x)x
> Calculate y’. sin (xy) = x2 - y
> Calculate y’. y = (1 – x-1)-1
> Find y’’ by implicit differentiation. x3 - y3 = 7
> Calculate y’. y = sec (1 + x2)
> Calculate y’. y = 3x ln x
> Calculate y’. y = ex sec x
> Calculate y’. y = cot (csc x)
> A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100 m ea
> Calculate y’. y + x cos y = x2y
> Calculate y’. y = ln sec x
> Calculate y’. y = (arcsin 2x)2
> Calculate y’. y = emx cos nx
> Calculate y’. y = ln (x ln x)
> Show that sin-1(tanh x) = tan-1(sinh x).
> 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 x-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.
> Find points P and Q on the parabola y == 1 2x2 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 С х
> Differentiate the function. y = x5/3 – x2/3
> A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surfa
> A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely
> A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 2/5 intersects some of these circles.
> Suppose that three points on the parabola y = x2 have the property that their normal lines intersect at a common point. Show that the sum of their x-coordinates is 0.
> Find the two points on the curve y = x4 - 2x2 - x that have a common tangent line.
> Given an ellipse x2/a2 + y2/b2 = 1, where a ≠ b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.
> For which positive numbers a is it true that ax ≥ 1 + x for all x?
> (a) The cubic function f (x) = x(x – 2)(x – 6) has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function f (x) = (x – a)(x – b)(x – c) has three distinct
> Find y’’ by implicit differentiation. x2 + xy + y2 = 3
> Suppose that we replace the parabolic mirror of Problem 22 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mir
> Let P(x1, y1) be a point on the parabola y2 = 4px with focus F(p, 0). Let α be the angle between the parabola and the line segment FP, and let β be the angle between the horizontal line y = y1 and the parabola as in the figure.
> (a) The curve with equation y2 = x3 + 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, -2). (b) At what points does this curve have horizontal tangents? (c) Illustrate parts (a) and (b) by graphin
> 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 yT 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
> Find y’’ by implicit differentiation. x2 + 4y2 = 4
> An equation of motion of the form s = Ae-ct cos( ωt + δ) represents damped oscillation of an object. Find the velocity and acceleration of the object.
> Find a parabola y = ax2 + bx + c that passes through the point (1, 4) and whose tangent lines at x = -1 and x = 5 have slopes 6 and -2, respectively.
> (a) Find an equation of the tangent to the curve y = ex that is parallel to the line x - 4y = 1. (b) Find an equation of the tangent to the curve y = ex that passes through the origin.
> At what point on the curve y = [ln(x + 4)]2 is the tangent horizontal?
> Show by implicit differentiation that the tangent to the ellipse x2 / a2 + y2 / b2 = 1 at the point (x0, y0) is x0x/a2 + y0y/b2 = 1
> (a) Graph the function f (x) = x - 2 sin x in the viewing rectangle [0, 8] by [-2, 8]. (b) On which interval is the average rate of change larger: [1, 2] or [2, 3]? (c) At which value of x is the instantaneous rate of change larger: x = 2 or x = 5? (d) C
> Find h’ in terms of f ’ and g’. h(x) = f (g(sin 4x))
> 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 x-axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity o
> Find f ‘ in terms of g’. f(x) = g(ln x)
> Find f ‘ in terms of g’. f(x) = eg(x)
> Find f ‘ in terms of g’. f(x) = g(ex)
> Find f ‘ in terms of g’. f(x) = g(g(x)
> Find f ‘ in terms of g’. f(x) = [g(x)]2
> Find the points on the lemniscate in Exercise 31 where the tangent is horizontal. Data from Exercise 31: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 2(x2 + y2)2 = 25(x2 - y2), (3, 1), (lemniscate
> Find f ‘ in terms of g’. f(x) = g(x2)
> Find f ‘ in terms of g’. f(x) = x2g(x)
> If f and t are the functions whose graphs are shown, let P(x) = f (x) g(x), Q(x) = f (x)/g(x), and C(x) = f (g(x)). Find (a) P’(2), (b) Q’(2), and (c) C’(2).
> Suppose that f (1) = 2 f ‘(1) = 3 f (2) = 1 f ‘(2) = 2 g(1) = 3 g’(1) = 1 tg(2) = 1 g’(2) = 4 (a) If S(x) = f (x) + g (x), find S’(1). (b) If P(x) = f (x) g(x), find P’(2). (c) If Q(x) = f(x)/g(x), find Q’(1). (d) If C(x) = f (g(x)), find C’(2).
> (a) By differentiating the double-angle formula cos 2x = cos2x - sin2x obtain the double-angle formula for the sine function. (b) By differentiating the addition formula sin(x + a) = sin x cos a + cos x sin a obtain the addition formula for the cosine fu
> Find the points on the ellipse x2 + 2y2 = 1 where the tangent line has slope 1.
> At what points on the curve y = sin x + cos x, 0 ≤ x ≤ 2π, is the tangent line horizontal?
> (a) If f (x) = 4x - tan x, -π/2 < x < π/2, find f ‘ and f ’’. (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f ‘, and f ’’.
> (a) The curve with equation 2y3 + y2 - y5 = x4 - 2x3 + x2 has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why. (b) At how many points does this curve have horizontal tangent lines? Find the x-coordinat
> If f (x) = xesin x, find f (x). Graph f and f ‘ on the same screen and comment.
> Find equations of the tangent line and normal line to the curve at the given point. y = (2 + x)e-x, (0, 2)
> How many lines are tangent to both of the circles x2 + y2 = 4 and x2 + (y – 3)2 = 1? At what points do these tangent lines touch the circles?
> A particle moves on a vertical line so that its coordinate at time t is y = t3 - 12t + 3, t ≥ 0. (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the parti
> Find the derivative. Simplify where possible. y = sech x (1 + ln sech x)
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4, ( -3√3 , 1) (astroid) yA 8
