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mathematics
calculus

Questions and Answers of Calculus

Find f'(x) If it is known that d/dx [f(2x)] = x2
If f(t) = √4t + 1. Find f"(2).
Find y" if x6 + y6 = 1.
Use mathematical induction (page 76) to show that if f(x) = xex, then f(n) (x) = (x + n)ex.
Find an equation of the tangent to the curve at the given point. (a) y = 4 sin2x, (π/6, 1) (b) y = √1 + 4 sin x, (0, 1)
If f(x) = xesin x, find f'(x). Graph f and f' on the same screen and comment.
(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
At what points on the curve y = sin x + cos x, 0 ≤ x ≤ 2π, is the tangent line horizontal?
If f(x) = (x - a) (x - b) (x - c), show that
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) (b) F'(2).
Find f' in terms of g'. (a) f(x) = x2 g(x) (b) f(X) = [g(x)]2 (c) f(x) = g(ex)
Find h' In terms of f' and g'. (a) h(x) = f(x) g(x)/f(x) + g(x) (b) h(x) = f(g (sin 4x)
At what point on the curve y = [In (x + 4)]2 is the tangent horizontal?
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.
An equation of motion of the form x = Ae-ct cos(wt + δ) represents damped oscillation of an object. Find the velocity and acceleration of the object.
A particle moves on a vertical line so that its coordinate at time is y = t3 - 12 t + 3, t ≥ 0. (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is
The mass of part of a wire is x(1 + √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 bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after
Let C(t) be the concentration of a drug in the bloodstream. As the body eliminates the drug, C(t) decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus
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?
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
Find points P and Q on the parabola y = 1 - x2 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.)
The figure shows a circle with radius 1 inscribed in the parabola y = x2. Find the center of the circle.
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 -axis as the wheel rotates counter clockwise at a rate of 360
Show that Dn/dxn(eax sin bx) = rn eax sin (bx + nθ) Where a and b are positive numbers, r2 = a2 + b2, and θ = tan-1(b/a).
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
(a) Use the identity for tan(x - y) (see Equation 14b in Appendix D) to show that if two lines L1 and L2 intersect at an angle α, then tan α = m2 - m1 / 1 + m1m2 where and are the slopes of and ,
Suppose that we replace the parabolic mirror of Problem 20 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
Evaluate sin(a + 2x) - 2 sin(a + x) + sin a/x2.
Evaluate sin(a + 2x) - 2 sin(a + x) + sin a/x2.
If y = x/√a2 - 1 - 2/√a2 - 1 arctan sin x/ a + √a2 - 1 + cos x show that y' = 1/a + cos x.
Find the two points on the curve y = x4 - 2x2 - x that have a common tangent line.
Show that the tangent lines to the parabola y = ax2 + bx + c at any two points with -coordinates p and q must intersect at a point whose -coordinate is halfway between p and q.
A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius are drawn using all lattice points as centers. Find the smallest value of such that any line with
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
If f(x) sec t - sec x/ t - x, find the value of f'(Ï€/4).
Show that sin-1 (tanh x) = tan-1 (sinh x).
Prove that dn/dxn (sin4 x + cos4 x) = 4n-1 cos(4x + nπ / 2).
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)
State the derivative of each function. (a) y = xn (b) y = ex (c) y = ax (d) y = In x (e) y = loga x (f) y = sin x (g) y = cos x (h) y = tan x (i) y = csc x (j) y = sec x (k) y = cot x (l) y = sin-1
(a) How is the number defined?(b) Express 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
(a) Explain how implicit differentiation works. (b) Explain how logarithmic differentiation works.
(a) Write a differential equation that expresses the law of natural growth. (b) Under what circumstances is this an appropriate model for population growth?
(a) Write an expression for the linearization of f at a. (b) If y = f(x), write an expression for the differential dy. (c) If dx = Δx, draw a picture showing the geometric meanings of Δy and dy.
Explain the difference between an absolute minimum and a local minimum.
(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable
(a) Sketch the graph of a function on [- 1, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [- 1, 2] that is discontinuous but has both an absolute
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) (a) f(x) = 1/2 (3x -
Find the critical numbers of the function. (a) f(x) = 4 + 1/3x - 1/2x2 (b) f(x) = 2x3 - 3x2 - 36x (c) g(t) = t4 + t3 + t2 + 1
A formula for the derivative of a function is given. How many critical numbers does have? f'(x) = 5e-0.1|x| sin x - 1
Find the absolute maximum and absolute minimum values of f on the given interval. (a) f(x) = 12 + 4x - x2, [0, 5] (b) f(x) = 2x3 - 3x2 - 12x + 1, [-2, 3] (c) f(x) = x3 - 6x2 + 5, [-3, 5]
Use the graph to state the absolute and local maximum and minimum values of the function.
If and are positive numbers, find the maximum value of f(x) = f(x) = xa(1 - x)b, 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
Between 0°C and30°C, the volume (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
Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. (a) Absolute minimum at 2, absolute maximum at 3, local minimum at 4 (b) Absolute maximum at 5, absolute
A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function S(t) = - 0.00003237t5 + 0.0009037t4 - 0.008956t3 + 0.03629t2 - 0.04458t + 0.4074 where is
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
Prove that the function F(x) = x101 + x51 + x + 1 has neither a local maximum nor a local minimum.
Prove Fermat's Theorem for the case in which f has a local minimum at c.
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolle's Theorem. (a) f(x) = 5 - 12x +
Find the number that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at (c, f(c)). Are the
Let f(x) = (x - 3)-2. Show that there is no value of in (1, 4) such that f(4) - f(1) = f'(c) (4 - 1). Why does this not contradict the Mean Value Theorem?
Show that the equation has exactly one real root. 2x + cos x = 0.
Show that the equation x3 - 15x + c = 0 has at most one root in the interval [-2, 2].
(a) Show that a polynomial of degree 3 has at most three real roots. (b) Show that a polynomial of degree has at most real roots.
If f(1) = 10 and f'(x) ≥ 2 for 1 ≤ x ≤ 4, how small can f(4) possibly be?
Does there exist a function f such that f(0) = - 1, f(2) = 4, and f'(x) ≤ 2for all x?
Show that √1 + x < 1 + 1/2 x if x > 0.
Use the Mean Value Theorem to prove the inequality |sin a - sin b| ≤ |a - b| for all a and b
Let f(x) = 1/x andShow that f'(x) = g'(x) for all x in their domain. Can we conclude from Corollary 7 that f - g is constant?
Prove the identity arcsin x - 1/x + 1 = 2 arctan √x - π/2
Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. Consider f(t) = g(t) - h(t), where g and h are the position function
Let f(x) = 1 - x2/3. Show that f(-1) = f(-1) = f(1) but there is no number c in (-1, 1) such that f'(c). Why does this not contradict Rolle's Theorem?
Use the graph of f to estimate the values of that satisfy the conclusion of the Mean Value Theorem for the interval [0, 8].
Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. (a) f(x) = 2x2 - 3x +
Use the given graph of to find the following.(a) The open intervals on which f is increasing.(b) The open intervals on which f is decreasing.(c) The open intervals on which f is concave upward.(d)
Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x) = 1 + 3x2 - 2x3
Suppose f" is continuous on (-∞, ∞). (a) If f'(2) = 0 and f"(2) = - 5, what can you say about f? (b) If f" (6) = 0 and f"(6) = 0, what can you say about f?
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) <
Suppose you are given a formula for a function f. (a) How do you determine where f is increasing or decreasing? (b) How do you determine where the graph of f is concave upward or concave
The graph of the derivative f' of a continuous function f is shown.(a) On what intervals is f increasing? Decreasing?(b) At what values of x does f have a local maximum? Local minimum?(c) On what
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
The graph of the derivative f of a function f is shown.(a) On what intervals is f increasing or decreasing?(b) At what values of x does f have a local maximum or minimum?
Suppose the derivative of a function f is f'(x) = (x + 1)2 (x - 3)5 (x - 6)4. On what interval is increasing?
(a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value. f(x) = x + 1/
(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of f" to give better estimates. F(x) = cos x + 1/2 cos 2x,
Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f". F(x) = x4 + x3 + 1 / √x2 + x + 1
A graph of a population of yeast cells in a new laboratory culture as a function of time is shown.(a) Describe how the rate of population increase varies.(b) When is this rate highest?(c) On what
Let k(t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, k(8) - K(7) or K(3) - K(2)? Is the graph of K concave upward or concave downward? Why?
A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = AtPe-kt is often used to model the response curve, reflecting an
Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of at x = - 2 and a local minimum value of 0 at x = 1.
(a) If the function f(x) = x3 + ax2 + bx has the local minimum value -2/3 √3 at 1/√3, what are the values of a and b?(b) Which of the tangent lines to the curve in part (a) has the smallest slope?
In each part state the -coordinates of the inflection points of f. Give reasons for your answers.(a) The curve is the graph of f.(b) The curve is the graph of f'.(c) The curve is the graph of f".
Show that the curve y = (1 + x)/(1 + x2) has three points of inflection and they all lie on one straight line.
Show that the inflection points of the curve y = x sin x lie on the curve y2(x2 + 4) = 4x2.
Assume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and are positive, increasing, concave upward functions on I, show that the product function
Show that tan x > x for 0 < x < π/2.
Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three -intercepts x1, x2, and x3, show that the -coordinate of the inflection point

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