Find h' in terms of f' and g'. h (x) = f (g (sin 4x)
> Explain what each of the following means and illustrate with a sketch. (a) lim f(x) = L (b) lim f(x) = L エ→ (c) lim f(r) = L エ→a (d) lim f(x) = 0 エ→ %3D (e) lim f(x) = L
> Find an equation of the tangent line to the curve y = x√x that is parallel to the line y = 1 + 3x.
> Find the limit. limr→9 √r/ (r -9)4
> Find the limit. limt→2 t2 – 4/t3 - 8
> Find the limit. limh→0 (h – 1)3 + 1/h
> Find the limit. limx→1 x2 – 9/x2 + 2x -3
> Find the limit. limx→-3 x2 – 9/x2 + 2x - 3
> Let C (t) be the total value of US currency (coins and banknotes) in circulation at time t. The table gives values of this function from 1980 to 2000, as of September 30, in billions of dollars. Interpret and estimate the value of C'(1990). t 1980 1
> Find the limit. limx→3 x2 -9/x2 + 2x -3
> The graph of f is shown. State, with reasons, the numbers at which f is not differentiable. y -1 4. 2.
> (a). If f (x) = √3 – 5x, use the definition of a derivative to find f'(x). (b). Find the domains of f and f'. (c). Graph f and f' on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable.
> Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. y.
> (a). In Section 2.8 we defined an antiderivative of f to be a function F such that F' = f. Try to guess a formula for an antiderivative of f (x) = x2. Then check your answer by differentiating it. How many antiderivatives does f have? (b). Find antideriv
> Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. y
> Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.
> Find a function f and a number a such that limh→0 (2+h)6 – 64/h = f'(a)
> (a). If f (x) = ex, estimate the value of f'(1) graphically and numerically. (b). Find an approximate equation of the tangent line to the curve y = e-x2 at the point where x = 1. (c). Illustrate part (b) by graphing the curve and the tangent line on the
> (a). Use the definition of a derivative to find f'(2), where f (x) = x3 - 2x. (b). Find an equation of the tangent line to the curve y = x3 – 2x at the point (2, 4). (c). Illustrate part (b) by graphing the curve and the tangent line on the same screen.
> Find the limit. limx→1 ex3-x
> For the function f whose graph is shown, arrange the following numbers in increasing order: 0 1 f'(2) f'(3) f'(5) f"(5) -1 1
> The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + 1/4t2, where is measured in seconds. (a). Find the average velocity over each time period. (b). Find the instantaneous velocity when t = 1. (i) [1, 3] (iii
> Prove that limx→0 x2 cos (1/x2) = 0.
> If 2x – 1 < f (x) < x2 for 0 < x < 3, find limx→1 f (x).
> When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. (a). Sketch a possible graph of T as a function of the time that has elapsed since the faucet was turned on. (b). Describe how the rate of
> Sketch the graph of an example of a function that satisfies all of the following conditions: f is continuous from the right at 3 lim f(x) = -2, lim f(x) = 0, lim f(x) = 0, %3D -3 lim f(x) = -0, lim f(x) = 2,
> Use graphs to discover the asymptotes of the curve. Then prove what you have discovered. y = cos2x/x2
> Find the limit. limx→1 (1/x – 1+ 1/x2 – 3x + 2)
> Find the limit. limx→∞ (√x2 + 4x + 1 - x)
> Find the limit. limx→∞ ex-x2
> Evaluate limx→0 ((√1 + tan x) – (√1 + sin x)/x3)
> Find f'(x) if it is known that d/dx [f (2x)] = x2
> 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 (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 = arctan (arcsin √x)
> 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
