Define the derivative f'(a). Discuss two ways of interpreting this number.
> Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. f(x) = Vx4 – 2x2 + 2 – 2, -3 <I<3
> Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. sin x f(x) = 1+ x?"
> The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.
> The graph of a function f is shown. Which graph is an antiderivative of f and why? a
> Let f (x) = x4 – 2x2. (a). Use the definition of a derivative to find f'(x) and f"(x). (b). On what intervals is f increasing or decreasing? (c). On what intervals is f concave upward or concave downward?
> If f'(x) = e-x2, what can you say about f?
> For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?
> Sketch the graph of a function that satisfies all of the given conditions. f'(x) > 0 if |x| < 2, f'(x) < 0 if |x| > 2, f'(-2) = 0, lim |f'(x) | = ∞, f"(x) > 0 if x # 2
> Sketch the graph of a function that satisfies all of the given conditions. f'(1) = f'(-1) = 0, f'(x) < 0 if |x| < 1, f'(x) > 0 if 1 < |x|< 2, f'(x) = -1 if |x|> 2, f"(x) < 0 if –2 <x< 0, inflection point (0, 1)
> Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(2) = f'(4) = 0, f'(x) > 0 if x < 0 or 2 <x< 4, f'(x) < 0 if 0 <x< 2 or x> 4, f"(x) > 0 if 1 < x< 3, f"(x) < 0 if x < 1 or x > 3
> Sketch the graph of a function that satisfies all of the given conditions. f'(x) > 0 for all x # 1, vertical asymptote x = 1, f"(x) > 0 if x < 1 or x> 3, f"(x) < 0 if 1 <x<3
> Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(4) = 0, f'(x) > 0 if x < 0, f'(x) < 0 if 0 < x < 4 or if x > 4, f"(x) > 0 if 2 < x< 4, f"(x) < O if x < 2 or x>4
> Sketch the graph of a function whose first derivative is always negative and whose second derivative is always positive.
> Sketch the graph of a function whose first and second derivatives are always negative.
> The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its derivatives.
> (a). Sketch a curve whose slope is always positive and increasing. (b). Sketch a curve whose slope is always positive and decreasing. (c). Give equations for curves with these properties.
> Let P (x1, y1) be a point on the parabola y2 = 4px with focus F (p, 0). Let a be the angle between the parabola and the line segment FP, and let β be the angle between the horizontal line y = y' and the parabola as in the figure. Prove that a
> Estimate the value of f'(a) by zooming in on the graph off. Then differentiate f to find the exact value of f'(a) and compare with your estimate. f (x) = 1/√x, a = 4
> If limx→a [f (x) + g (x)] = 2 and limx→a [f (x) - g (x)] = 1, find limx→a [f (x) g (x)].
> If f is a differentiable function and g (x) = x f (x), use the definition of a derivative to show that g'(x) = x f'(x) + f (x).
> For what value of does the equation e2x = k√x have exactly one solution?
> If y = ((x/√a2 – 1) – (2/√a2 – 1) arctan sin x/a + (√a2 – 1) + cos x) show that y' = 1/a + cos x.
> If f and g are differentiable functions with f (0) = g (0) = 0 and g'(0) ≠0, show that f(x) _ f'(0) lim g(x) g'(0)
> If [[x]] denotes the greatest integer function, find limx→∞ x/[[x]].
> Find the values of the constants a and b such that limx→0 3√ax + b – 2/x = 5/12
> Show that dn/dxn = eax sin bx) = rneaxs sin (bx + nθ) where a and b are positive numbers, r2 = a2 + b2, and θ = tan-1(b/a).
> Tangent lines T1 and T2 are drawn at two points P1 and P2 on the parabola y = x2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2; it intersects T1 at Q1 and T2 at Q2. Show that | PQ.| | P | PQ:| 1
> Find all values of a such that f is continuous on R: Jx +1 if x< a f(x) = |x? - if x>a
> Find a cubic function y = ax2 + bx2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2, 6).
> If f is differentiable at a, where a > 0, evaluate the following limit in terms of f'(a): f(x) – f(a) lim VI - Va x,
> If f (x) = limt→x sec t – sec x/t - x, find the value of f'(π/4).
> Show that d/dx (sin2 x/ 1 + cot x + cos2x/1 + tan x) = -cos 2x
> Find numbers a and b such that limx→1 √ax + b – 2/x = 1
> Evaluate limx→1 3√x – 1/√x - 1
> Bézier curves are used in computer-aided design and are named after the French mathematician Pierre Bézier (1910–1999), who worked in the automotive industry. A cubic Bézier curve is determined by
> The tangent line approximation L (x) is the best first-degree (linear) approximation f (x) to near x = a because f (x) and L (x) have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a s
> Certain combinations of the exponential functions, ex and e-x arise so frequently in mathematics and its applications that they deserve to be given special names. This project explores the properties of functions called hyperbolic functions. The hyperbol
> State each differentiation rule both in symbols and in words. (a). The Power Rule (b). The Constant Multiple Rule (c). The Sum Rule (d). The Difference Rule (e). The Product Rule (f). The Quotient Rule (g). The Chain Rule
> (a). Suppose the horizontal distance between P and Q is 100 ft. Write equations in a, b, and that will ensure that the track is smooth at the transition points. (b). Solve the equations in part (a) for a, b, and c to find a formula for f (x). (c). Plot L
> Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 – 3x. Find the values of a, b, c, and d.
> The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L1 (x) for x 100] doesn’t have a continuous second derivative. So you decide to improve the design by using
> An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: (i). The cruising altitude is h when descent starts at a horizontal distance from touchdown at the origin. (ii). The pilot must maintain a constant
> (a). What does it mean to say that the line x = a is a vertical asymptote of the curve y = f (x)? Draw curves to illustrate the various possibilities. (b). What does it mean to say that the line y = L is a horizontal asymptote of the curve y = f (x)? Dra
> Describe several ways in which a limit can fail to exist. Illustrate with sketches.
> (a). Define an antiderivative of f. (b). What is the antiderivative of a velocity function? What is the antiderivative of an acceleration function?
> (a). What does the sign of f'(x) tell us about f? (b). What does the sign of f"(x) tell us about f?
> Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.
> (a). What does it mean for f to be differentiable at a? (b). What is the relation between the differentiability and continuity of a function? (c). Sketch the graph of a function that is continuous but not differentiable at a =2.
> Define the second derivative of f. If f (t) is the position function of a particle, how can you interpret the second derivative?
> Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x - 2.
> If y = f (x) and x changes from x1 to x2, write expressions for the following. (a). The average rate of change of y with respect to x over the interval [x1, x2]. (b). The instantaneous rate of change of y with respect to x at x = x1.
> State the following Limit Laws. (a). Sum Law (b). Difference Law (c). Constant Multiple Law (d). Product Law (e). Quotient Law (f). Power Law (g). Root Law
> State the derivative of each function. (a). y = xn (b). y = ex (c). y = ax (d). y = ln x (e). y = logax (f). y = sin x (g). y = cos x (h). y = tan x (i). y = y = csc x (j). y = sec x (k). y = cot x (l). y = sin-1 x (m). y = cos-1 x (n). y = tan-1 x
> Which of the following curves have vertical asymptotes? Which have horizontal asymptotes? (a). y = x4 (b). y = sin x (c). y = tan x (d). y = ex (e). y = ln x (f). y = 1/x (g). y = √x
> (a). What does it mean for f to be continuous at a? (b). What does it mean for f to be continuous on the interval (∞,∞)? What can you say about the graph of such a function?
> Write an expression for the linearization of, f at a.
> How do you find the slope of a tangent line to a parametric curve x = f (t), y = g (t)?
> (a). Explain how implicit differentiation works. When should you use it? (b). Explain how logarithmic differentiation works. When should you use it?
> (a). How is the number e defined? (b). Express e as a limit. (c). Why is the natural exponential function y = ex used more often in calculus than the other exponential functions y = ax? (d). Why is the natural logarithmic function y = ln x used more oft
> 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.
