Calculate y'. y = arctan (arcsin √x)
> Express the limit limθ→π/3 cos θ – 0.5/θ – π/3 as a derivative and thus evaluate it.
> A window has the shape of a square surmounted by a semicircle. 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 the
> (a). Find the linearization of f (x) = 3√1 + 3x at a = 0. State the corresponding linear approximation and use it to give an approximate value for 3√1.03. (b). Determine the values of for which the linear approximation given in part (a) is accurate to wi
> Let f (x) = 3√x. (a). If a ≠ 0, use Equation 2.6.5 to find f'(a). (b). Show that f'(a) does not exist. (c). Show that y = 3√x has a vertical tangent line at (0, 0).
> The function C (t) = K (e-at – e-ht), where a, b, and K are positive constants and b > a, is used to model the concentration at time t of a drug injected into the bloodstream. (a). Show that limt→∞ C (t) = 0. (b). Find C"(t), the rate at which the drug i
> Find the limit. limx→∞ √x2 – 9/2x - 6
> The volume of a right circular cone is V = 1/3πr2h, where r is the radius of the base and h is the height. (a). Find the rate of change of the volume with respect to the height if the radius is constant. (b). Find the rate of change of the volume with re
> The mass of part of a wire is x (x + √x) kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x = 4 m.
> A particle moves along a horizontal line so that its coordinate at time t is x = √b2 + c2t2, t > 0, where and are positive constants. (a). Find the velocity and acceleration functions. (b). Show that the particle always moves in the positive direction.
> 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 the limit. limx→-∞ 1 – 2x4 - x4/5 + x – 3x4
> Find the limit. limx→π- ln (sinx)
> (a). On what interval is the function f (x) = (ln x)/x increasing? (b). On what interval f is concave upward?
> Find the points on the ellipse x2 + 2y2 = 1 where the tangent line has slope 1.
> Find equations of both lines that are tangent to the curve y = 1 + x3 and parallel to the line 12x - y = 1.
> Graphs of the position functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) SA (b) SA to ++ 1
> Find the limit. limx→3 √x + 6 – x/x3 - 3x2
> At what point on the curve y = [ln (x + 4)]2 is the tangent horizontal?
> Find h' in terms of f' and g'. h (x) = f (g (sin 4x)
> Find h' in terms of f' and g'. h (x) = f (x) g (x)/f (x) + g (x)
> Find f' in terms of g'. f (x) = g (ln x)
> Find f' in terms of g'. f (x) = ln |g (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 expressions for the first five derivatives of f (x) = x2ex. Do you see a pattern in these expressions? Guess a formula for f(n)(x) and prove it using mathematical induction.
> Find f' in terms of g'. f (x) = g (x2)
> Find f' in terms of g'. f (x) = x2g (x)
> If f and g 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). 9/ f
> Suppose that h (x) f (x) g (x) and F (x) = f (g (x)), where f (2) = 3, g (2) = 5, g'(2) = 4, f'(2) = -2, and f'(5) = 11. Find (a) h'(2) and (b) F'(2).
> (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] 0r [2, 3]. (c). At which value of is the instantaneous rate of change larger: x = 2 or x = 5? (d).
> If f (x) =xesin x, find f'(x). Graph f and f' on the same screen and comment.
> Find the limit. limu→1 u4 – 1/u3 + 5u2 – 6u
> Find the limit. limv→14+ 4 – v/|4 – v|
> Find equations of the tangent line and normal line to the curve at the given point. x2 + 4xy + y2 = 13, (2, 1)
> Sketch the graph of a function that satisfies the given conditions: f(0) = 0, f'(-2) =f'(1) = f"(9) = 0, lim f(x) = 0, lim f(x) = -, %3D %3D f'(x) < 0 on (-0, -2), (1, 6), and (9, 0), f'(x) > 0 on (-2, 1) and (6, 9), f"(x) > 0 on (-0, 0) and (12, 0)
> (a). Find equations of both lines through the point (2, -3) that are tangent to the parabola y = x2 + x. (b). Show that there is no line through the point (2, 7) that is tangent to the parabola. Then draw a diagram to see why.
> The figure shows the graph of the derivative f' of a function f. (a). Sketch the graph of f". (b). Sketch a possible graph of f. y. y = f'(x) -1 1 2 3 /4 5 6 7
> Find an equation of the tangent to the curve at the given point. x = ln t, y = t2 + 1, (0, 2)
> Find an equation of the tangent to the curve at the given point. y = x2 - 1/x2 + 1, (0, -1)
> Find an equation of the tangent to the curve at the given point. y = 4 sin2x, (π/6, 1)
> Find y" if x6 + y6 = 1.
> If f (x) = 2x, find f(n)(x).
> If g (θ) = θ sin θ, find g"(π/6).
> If f (t) = √4t + 1, find f"(2).
> Calculate y'. y = sin2 (cos √sin πx)
> Calculate y'. y = cos (e√tan 3x)
> Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the point (0, -4). Find the coordinates of the points where these tangent lines intersect the parabola.
> Calculate y'. y = sin (tan √1 + x3)
> Calculate y'. y = ln |x2 – 4/2x + 5|
> Calculate y'. y = tan2(sin θ)
> Calculate y'. y = 10 tan πθ
> Calculate y'. y = ln | sec 5x + tan 5x|
> Calculate y'. y = ecos x + cos (ex)
> Calculate y'. y = x tan-1(4x)
> Calculate y'. y = (x2 + 1)4/) (2x + 1)3(3x – 1)5
> Calculate y'. y = ln sin x – ½ sin2x
> The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 -5 3 x² + 4y? = 5
> Calculate y'. xey = y - 1
> Calculate y'. y = 3x ln x
> Calculate y'. y = √t ln (t4)
> Calculate y'. sin (xy) = x2 - y
> Calculate y'. y = (ln x) cos x
> Calculate y'. y = log5 (1 + 2x)
> Calculate y'. y = ln (x2ex)
> Calculate y'. y = ecx (c sin x – cos x)
> Calculate y'. x2 cos y + sin 2y = xy
> Calculate y'. y = sec 2θ/1 + tan 2θ
> If f and g are the functions whose graphs are shown, let u (x) = f (g (x)), v (x) = g (f (x)), and w (x) = g (g (x)). Find each derivative, if it exists. If it does not exist, explain why. (a). u'(1) (b). v'(1) (c). w'(1) y f
> Calculate y'. y = ln (csc 5x)
> Calculate y'. xy4 + x2y = x + 3y
> Calculate y'. y = (arcsin 2x)2
> Calculate y'. y = ((e1/x)/x2)
> Calculate y'. y = emx cos nx
> Calculate y'. y = t/1 – t2
> Calculate y'. y = e-1(t2 – 2t + 2)
> Calculate y'. y = esin2θ
> Calculate y'. y = ex/1 + x2
> Calculate y'. y = 2x√x2 + 1
> Find equations of the tangent lines to the curve y = x – 1/x + 1 that are parallel to the line x - 2y = 2.
> Calculate y'. y = 3x – 2/√2x + 1
> Calculate y'. y = √x + 1/3√x4
> Calculate y'. y = cos (tan x)
> Calculate y'. y = (x4 – 3x2 + 5)3
> Show that the length of the portion of any tangent line to the asteroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is constant.
> The cost, in dollars, of producing 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. (d).
> 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 pa
> 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.
> The graph of a function is shown. Sketch the graph of an antiderivative F, given that F (0) = 0.
> Recall that a function f is called even if f (-x) = f (x) for all x in its domain and odd if f (-x) = -f (x) for all such x. Prove each of the following. (a). The derivative of an even function is an odd function. (b). The derivative of an odd function i
> A car starts from rest and its distance traveled is recorded in the table in 2-second intervals. (a). Estimate the speed after 6 seconds. (b). Estimate the coordinates of the inflection point of the graph of the position function. (c). What is the sign
> (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). If f (x) = x√5 - x, find f'(x). (b). Find equations of the tangent lines to the curve y = x√5 - x at the points (1, 2) and (4, 4). (c). Illustrate part (b) by graphing the curve and tangent lines on the same screen. (d). Check to see that your answe
> The total fertility rate at time t, denoted by F (t), is an estimate of the average number of children born to each woman (assuming that current birth rates remain constant). The graph of the total fertility rate in the United States shows the fluctuatio
> Find equations of the tangent line and normal line to the curve at the given point. y = (2 + x) e-x, (0, 2)
> Find an equation of the tangent to the curve at the given point. x = t3 – 2t2 + t + 1, y = t2 + t, (1, 0)
> The graph of the derivative f' of a function f is given. (a). On what intervals is f increasing or decreasing? (b). At what values of x does f have a local maximum or minimum? (c). Where is f concave upward or downward? (d). If f (0) = 0, sketch a poss
> The cost of living continues to rise, but at a slower rate. In terms of a function and its derivatives, what does this statement mean?
