(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 ’’.
> 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.
> 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) 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
> 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
> Prove Equation 4. Equation 4: y = In(x + væ² – 1).
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + y2 = (2x2 + 2y2 – x)2, (0, 1/2), (cardioid) y.
> Find equations of the tangent lines to the curve y = x – 1 / x + 1 that are parallel to the line x - 2y = 2.
> Give an alternative solution to Example 3 by letting y = sinh-1x and then using Exercise 9 and Example 1(a) with x replaced by y. Example 3: Exercise 9: Example 1(a): / Show that sinhx = In(x + /x? + 1). cosh x + sinh x = e'
> Prove the formulas given in Table 1 for the derivatives of the functions (a) cosh, (b) tanh, (c) csch, (d) sech, and (e) coth. Table 1: Table 1 N as a function of t N=f(t) = population at time t || (hours) 100 1 168 2 259 3 358 4 445 5 509 550 7 57
> (a) Use the graphs of sinh, cosh, and tanh in Figures 1–3 to draw the graphs of csch, sech, and coth. (b) Check the graphs that you sketched in part (a) by using a graphing device to produce them. Figures 1 -3: y. y= y= sinh x `
> If cosh x = 5/3 and x > 0, find the values of the other hyperbolic functions at x.
> If tanh x = 12 / 13, find the values of the other hyperbolic functions at x.
> Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?
> Prove the identity. (cosh x + sinh x)n = cosh nx 1 sinh nx (n any real number)
> Prove the identity. cosh 2x = cosh2x + sinh2x
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + 2xy + 4y2 = 12, (2, 1) (ellipse)
> Prove the identity. sinh 2x = 2 sinh x cosh x
> Prove the identity. coth2x - 1 = csch2x
> Prove the identity. cosh(x + y) = cosh x cosh y + sinh x sinh y
> Prove the identity. Sinh(x + y) = sinh x cosh y + cosh x sinh y
> A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 308. At what rate is the distance from the plane to the radar station increasing a minute later?
> Prove the identity. cosh x - sinh x −= e-x
> Prove the identity. cosh x + sinh x = ex
> Prove the identity. cosh(-x) = cosh x (This shows that cosh is an even function.)
> Prove the identity. sinh(-x) = -sinh x (This shows that sinh is an odd function.)
> Find the numerical value of each expression. (a) sinh 1 (b) sinh-1 1
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 - xy - y2 = 1, (2, 1) (hyperbola)
> Find the linearization L(x) of the function at a. f (x) = 2x, a = 0
> Find the linearization L(x) of the function at a. f (x) = x , a = 4
> Find the linearization L(x) of the function at a. f (x) = sin x, a = π/6
> Find the linearization L(x) of the function at a. f (x) = x3 - x2 + 3, a = -2
> A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?
> Suppose that the only information we have about a function f is that f (1) = 5 and the graph of its derivative is as shown. (a) Use a linear approximation to estimate f (0.9) and f (1.1). (b) Are your estimates in part (a) too large or too small? Explai
> 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
> If a current I passes through a resistor with resistance R, Ohm’s Law states that the voltage drop is V = RI. If V is constant and R is measured with a certain error, use differentials to show that the relative error in calculating I is approximately the
> 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
> One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30°, with a possible error of 61°. (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 Dr. (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.
> A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time?
