Find y’’ if x6 + y6 = 1.
> Find an equation of the tangent line to the hyperbola x2 / a2 - y2 / b2 = 1 at the point (x0, y0).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. An equation of the tangent line to the parabola y = x2 at (-2, 4) is y - 4 = 2x(x + 2).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a rational function is a rational function.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. . If f (x) = (x6 – x4)5, then f (31)(x) = 0.
> The cost, in dollars, of producing x units of a certain commodity is C(x) = 920 + 2x - 0.02x2 + 0.00007x3 (a) Find the marginal cost function. (b) Find C’(100) and explain its meaning. (c) Compare C’(100) with the cost of producing the 101st item.
> Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a polynomial is a polynomial.
> Show that the length of the portion of any tangent line to the astroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is constant.
> The Michaelis-Menten equation for the enzyme chymotrypsin is v = 0.14[S] / 0.015 + [S] where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dv/d[S] and interpret it.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If y = e2, then y’ = 2e.
> Suppose f is a differentiable function such that f (g(x)) = x and f ‘(x) = 1 + [f(x)]2 . Show that g’(x) = 1 / (1 + x2).
> Express the limit as a derivative and evaluate. lim ℎ→0 4 16+ℎ −2 ℎ
> A window has the shape of a square surmounted by a semi- circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of th
> Evaluate dy if y = x3 - 2x2 + 1, x = 2, and dx = 0.2.
> The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is π/6?
> Prove that the function f (x) = x101 + x51 + x + 1 has neither a local maximum nor a local minimum.
> A waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp? 4 ft -15 ft–
> Show that 5 is a critical number of the function t(x) = 2 + (x – 5)3 but t does not have a local extreme value at 5.
> 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
> Find the derivative of the function. Simplify where possible. H(t) = cot-1(t) + cot-1(1/t)
> 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
> The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function L(t) = 0.01441t3 - 0.4177t2 + 2.703t + 1060.1 where t is measured in months since January 1, 2012. Estimate when the water
> Between 0°C and 30°C, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula V = 999.87 - 0.06426T + 0.0085043T2 - 0.0000679T3 Find the temperature at which water has its maximum density.
> After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C(t) = 8(e-0.4t – e-0.6t) where the time t is measured in hours and C is measured in mg/mL. What is the maximum concentration of the an
> After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function C(t) = 1.35
> (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) = x - 2 cos x, -2 ≤ x ≤ 0
> A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?
> (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) = ex + e-2x, 0 ≤ 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).
> 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
> 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.
> 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.
