2.99 See Answer

Question: Differentiate the function. y = x5/3 – x2


Differentiate the function.
y = x5/3 – x2/3


> 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)

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> 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

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> 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

> Find y’’ by implicit differentiation. sin y + cos x = 1

> 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.

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> 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.

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> 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

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> 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

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> (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

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> Find f ‘ in terms of g’. f(x) = eg(x)

> Find f ‘ in terms of g’. f(x) = g(ex)

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> Find f ‘ in terms of g’. f(x) = x2g(x)

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> 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

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> 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

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> Find the derivative. Simplify where possible. G(t) = sinh (ln t)

> Find the derivative. Simplify where possible. F(t) = ln (sinh t)

> Find the derivative. Simplify where possible. h(x)= sinh (x2)

> Find the derivative. Simplify where possible. g(x) = sinh2 x

> Find the derivative. Simplify where possible. f(x) = ex cosh x

> Prove the formulas given in Table 6 for the derivatives of the following functions. (a) cosh-1 (b) tanh-1 (c) csch-1 (d) sech-1 (e) coth-1 Table 6: 6 Derivatives of Inverse Hyperbolic Functions d - (sinh¯'x) dx d -(csch-'x) = dx 1 1 1 + x? |x|/x? +

> For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3. (a) csch-1 (b) sech-1 (c) coth-1

> Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. x4y2 - x3y + 2xy3 = 0

2.99

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