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mathematics
calculus

Questions and Answers of Calculus

From area (OBP) ‰¤ area (sector OBP) ‰¤ area (OBP) + area (ABPQ) in Figure, show thatCos t ‰¤ t/ sin t ‰¤ 2 - cos tAnd thus obtain another proof that
In Figure 5, let D be the area of triangle ABP and E the area of the shaded region.(a) Guess the value ofBy looking at the figure. (b) Find a formula for D/E in terms of t. (c) Use a calculator to
In problem 1-5, find the limits.1.2. 3. 4. 5.
In problem 1-3, find the horizontal and vertical asymptotes for the graph of the indicated functions. Then sketch their graphs. 1. f(x) = 3/x + 1 2. f(x) = 3/(x + 1)2 3. f(x) = 2x /x - 3
The line y = ax + b s called an oblique asymptote to the graph of y = f(x) if eitheror Find the oblique asymptote for F(x) = 2x4 + 3x3 - 2x - 4 / x3 - 1
Find the oblique asymptote for F(x) = 3x3 + 4x2 - x + 1 / x2 + 1
Using the smbol M and Î´, give precise definition of each expressions.
Using the symbol M and N, give precise definitions of each expression.
Give a rigorous proof that ifAnd Then
Use a computer or a graphing calculator to find the limits in Problems 1-3. Begin by plotting the function in an appropriate window.1.2. 3.
Find the one-sided limits in Problem 1-3. Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to
In problem 1-4, state whether the indicated function is continuous at 3. If it is not continuous, tell why. 1. f(x) = (x - 3) (x - 4) 2. g(x) = x2 - 9 3. h(x) = 3/x - 3 4. g(t) = √t - 4
1. From the graph of g(see Figure 13), indicate the values where g is discontinuous. For each of these values state whether g is continuous from the right, left, or neither.2. From the graph of h
In problems 1-3, the given function is not defined at a certain point. How should it be defined in order to make it continuous at that points? (see example 1.) 1. f(x) = x2 - 49/x - 7 2. f(x) = 2x2 -
In problem 1-5, at what point, if any, are the functions discontinuous? 1. f(x) = 3x + 7 / (x - 30) (x - π) 2. f(x) = 33 - x2 / xπ + 3x - 3π - x2 3. h(θ) = |sin θ + cos θ| 4. r(θ) = tan θ 5.
Sketch the graph of a function f that satisfies all the following conditions. (a) Its domain is [-2, 2]. (b) f(-2) = f(-1) = f(1) = f(2) = 1. (c) It is discontinuous at - 1 and 1. (d) It is right
Sketch the graph of a function that has domain [0, 2] and is continuous on [0, 2) but not on [0, 2].
Sketch the graph of a function that has domain [0, 6] and is continuous on [0, 2] and (2, 6] but is not continuous on [0, 6].
Sketch the graph of a function that has domain [0, 6] and is continuous on (0, 6) but not on [0, 6].
In problems 1-5 determine whether the function is continuous at the given point c. If the function is not continuous, determine whether the discontinuity is removable or nonremovable.1. f(x) = sin x;
A cell phone company charges $0.12 for connecting a call plus $0.08 per minute or any part thereof (e.g., a phone call lasting 2 minutes and 5 seconds costs $0.12 + 3 × $0.08). Sketch a graph of the
A rental car company charges $20 for one day, allowing up to 200 miles. For each additional 100 miles, or any fraction thereof, the company charges $18. Sketch a graph of the cost for renting a car
A cab company charges $2.50 for the first 1/4 mile and $0.20 for each additional 1/8 mile. Sketch a graph of the cost of a cab ride as a function of the number of miles driven. Discuss the continuity
Use the Intermediate Value Theorem to prove that x3 + 3x - 2 = 0 has a real solution between 0 and 1.
Use the Intermediate Value Theorem to prove that (cos t)t3 + 6 sin5 t - 3 = 0 has a real solution between 0 and 2π.
Use the Intermediate Value Theorem to show that x3 - 7x2 + 14x - 8 = 0 has at least one solution in the interval [0, 5]. Sketch the graph of y = x3 - 7x2 + 14x - 8 over [0, 5]. How many solutions
Use the Intermediate Value Theorem to show that √x - cos x = 0 has a solution between 0 and π/2. Zoom in on the graph of y = - cos x to find an interval having length 0.1 that contains this
Show that the equation x3 + 4x3 - 7x + 14 = 0 has at least one real solution.
Prove that f is continuous at c if and only if limt→0 (c + t) = f(c).
Prove that if f is continuous at c and f(c) > 0 there is an interval (c - 6, c + 6) such that f(x) > 0 on this interval.
Prove that if f is continuous on [0, 1] and satisfies 0 s f(x) ≤ 1 there, then f has a fixed point that is, there is a number c in [0, 1] such that f(c) = c.
Find the values of a and b so that the following function is continuous everywhere.
A stretched elastic string covers the interval, [0, 1]. The ends are released and the string contracts so that it covers the interval [a, b], a ≥ b ≥ 1. Prove that his results in at least one
Let f(x) = 1/x-1. Then f(-2) = - 1/3 and f(2) = 1. Does the Intermediate Value Theorem imply the existence of a number c between - 2 and 2 such that f(c) = 0? Explain.
Starting at 4 A.M., a hiker slowly climbed to the top of a mountain, arriving at noon. The next day, he returned along the same path, starting at 5 A.M. and getting to the bottom at 11 A.M. Show that
Let D be a bounded, but otherwise arbitrary, region in the first quadrant. Given an angle Î¸. 0 ‰¤ Î¸ ‰¤ Ï€/2, D can be circum-scribed by a
Let f(x + y) = f(x) + f(y) for all x and y and suppose that f is continuous at x = 0. (a) Prove that f is continuous everywhere. (b) Prove that there is a constant in such that f (t) = mt for all t
Prove that if f (x) is a continuous function on an interval then so is the function |f (x)| = √(f (x))2.
Let f(x) = 0 if x is irrational and let f (x) = 1/q if x is the rational number p/ q in reduced form (q > 0). (a) Sketch (as best you can) the graph off on (0, 1). (b) Show that f is continuous at
A thin equilateral triangular block of side length 1 unit has its face in the vertical xy-plane with a vertex Vat the origin. Under the influence of gravity, it will rotate about V until a side hits
In problems 1-5, find the indicated limit or state that it does not exist.1.2. 3. 4. 5.
Prove using Îµ - Î´ argument that
Refer to f of Problem 24. (a) What are the values of x at which f is discontinuous? (b) How should f be defined at x = - 1 to make it continuous there?
Sketch the graph of a function f that satisfies all the following conditions.(a) Its domain is [0, 6].(b) f(0) = f(2) = f(4) = f(6) = 2.(c) f is continuous except at x = 2.
Use the intermediate Value Theorem to prove that the equation x5 - 4x3 - 3x + 1 = 0 has at least one solution between x = 2 and x = 3.
In problem 1-3, find the equations of all vertical and horizontal asymptotes for the given function. 1. f(x) = x/x2 +1 2. g(x) = x2 / x2 + 1 3. F(x) = x2 / x2 - 1
Let f(x) = x2. Find and simplify each of the following.(a) f(2)(b) f(2.1)(c) f(2.1) - f(2)(d) f(2.1) - f(2) / 2.1 - 2(e) f(a + h)(f) f(a + h) - f(a)(g) f(a + h) - f(a)/(a + h) - a(h)
Assume that a soap bubble retains its spherical shape as it expands. At time t = 0 the soap bubble has radius 2 centimeters. At time t = 1, the radius has increased to 2.5 centimeters. How much has
One airplane leaves an airport at noon flying north at 300 miles per hour. Another leaves the same airport one hour later and flies east at 400 miles per hour. (a) What are the positions of the
Repeat (a) through (h) of problem 1 for the function f(x) = 1/x.(a) f(2)(b) f(2.1)(c) f(2.1) - f(2)(d) f(2.1) - f(2) / 2.1 - 2(e) f(a + h)(f) f(a + h) - f(a)(g) f(a + h) - f(a)/(a + h) - a(h)
Repeat (a) through (h) of problem 1 for the function f(x) = ˆšx.(a) f(2)(b) f(2.1)(c) f(2.1) - f(2)(d) f(2.1) - f(2) / 2.1 - 2(e) f(a + h)(f) f(a + h) - f(a)(g) f(a + h) - f(a)/(a + h) -
Repeat (a) through (h) of Problem 1 for the function f(x) = x3 + 1.(a) f(2)(b) f(2.1)(c) f(2.1) - f(2)(d) f(2.1) - f(2) / 2.1 - 2(e) f(a + h)(f) f(a + h) - f(a)(g) f(a + h) - f(a)/(a + h) - a(h)
Write the first two terms in the expansions of the following: (a) (a + b)3 (b) (a + b)4 (c) (a + b)5
A wheel centered at the origin and of radius 10 centimeters is rotating counter-clock-wise at a rate of 4 revolutions per second. A point P on the rim of the wheel is at position (10.0) at time t =
The given limit is a derivative, but of what function f and at what point?(a)(b) (c) (d) (e) (f) (g) (h)
In Problem 1-3, assume that the function given are differentiable, and find the indicated derivative. 1. f'(t) if f(t) = h(g(t)) + g2(t) 2. G"(x) if G(x) = F(r(x) + s(x)) + s(x) 3. If f(x) = Q
Find the coordinates of the point on the curve y = (x - 2)2 where there is a tangent line that is perpendicular to the line 2x - y + 2 = 0.
A spherical balloon is expanding from the sun's heat. Find the rate of change of the volume of the balloon with respect to its radius when the radius is 5 meters.
If the volume of the balloon of Problem 35 is increasing at a constant rate of 10 cubic meters per hour, how fast is its radius increasing when the radius is 5 meters?
A trough 12 feet long has a cross section in the form of an isosceles triangle (with base at the top) 4 feet deep and 6 feet across. If water is filling the trough at the rate of 9 cubic feet per
An object is projected directly upward from the ground with an initial velocity of 128 feet per second. Its height s at the end of /seconds is s = 1281 - 16t2 feet. (a) When does it reach its maximum
An object moves on a horizontal coordinate line. Its directed distance s from the origin at the end of t seconds is s = t3 - 6t2 + 9t feet. (a) When is the object moving to the left? (b) What is its
Use the sketch of s = f(t) on Figure 1 to approximate each of the following.(a) f'(2)(b) f'(6)(c) Vavg on [3, 7](d) d/dt f(t2) at t = 2(e) d/dt [f2(t)] at t = 2(f) d/dt (f(f(t))) at t= 2
Find Dx30y in each case, (a) y = x19 + x12 + x5 + 10 (b) y = x20 + x19 + x18 (c) y = 7x21 + 3x20 (d) y = sin x + cos x (e) y = sin 2x (f) y = 1/x
Find dy/dx in each case. (a) (x - 1)2 + y2 = 5. (b) xy2 + yx2 = 1 (c) x3 + y3 = x3y3 (d) x sin (xy) = x2 + 1 (e) x tan (xy) = 2
Show that the tangent lines to the curves y2 = 4x3 and 2x2 + 3y2 = 14 at (1, 2) are perpendicular to each other.
Let y = sin(πx) + x2. If x change from 2 to 2.01, approximately how much does y change?
Let xy2 + 2y(x + 2)2 + 2 = 0. (a) If x changes from - 2.00 to - 2.01 and y > 0, approximately how much does y change? (b) If x changes from -2.00 to - 2.01 and y < 0, approximately how much does y
Suppose that f(2) = 3, f'(2) = 4, f"(2) = - 1, g(2) = 2, and e(2) = 5. Find each value. (a) d/dx [f2(x) + g3(x)] at x = 2 (b) d/dx [f(x) g(x)] at x = 2 (c) d/dx [f(g(x))] at x = 2 (d) D2x[f2(x) ] at
A 13-foot ladder is leaning against a vertical wall. If the bottom of the ladder is being pulled away from the wall at a constant rate of 2 feet per second how fast is the top end of the ladder
An airplane is climbing at a 15° angle to the horizontal. How fast is it gaining altitude if its speed is 400 miles per hour?
Given that Dx|x| = |x|/x, x ≠ 0, find a formula for (a) Dx(|x|2) (b) D2x|x| (c) D3x|x| (d) D2x(|x|2)
Given that Dt|t| = |t|/t, x ≠ 0, find a formula for (a) Dθ |sin θ| (b) Dθ |cos θ|
In Problems 1-3, find the indicated derivative by using the rules that we have developed. 1. Dx(3x5) 2. Dx(x3 - 3x2 + x-2) 3. Dz(z3 + 4z2 + 2z)
Find the linear approximation to the following functions at the given points. (a) √x + 1 at a = 3 (b) x cos x at a = 1
In problem 1-2, solve the given inequalities. (see Section 0.2.) 1. (x - 2) (x - 3) < 0 2. x2 - x - 6 > 0
Find all points on the graph y = tan2 x where the tangent line is horizontal.
Find all points on the graph of y = x + sin x where the tangent line is horizontal.
Find all points on the graph of y = x + sin x where the tangent line us parallel to the line y = 2 + x.
Andy wants to cross a river that is 1 kilometer wide and get to a point 4 kilometers downstream. (See Figure 2.) He can swim at 4 kilometers per hour and run 10 kilometers per hour. Assuming that he
Let f(x) = x - cos x. (a) Does the equation x - cos x = 0 have a solution between x = 0 and x = π? How do you know? (b) Find the equation of the tangent line at x = π/2. (c) Where does the tangent
Find a function whose derivative is (a) 2x (b) sin x (c) x2 + x + 1
Add 1 to each answer from Problem 21. Are these functions also solutions to Problem 21? Explain.
In problems 1-3, find the derivative f'(x) of the given function. 1. f(x) = (2x + 1)4 2. f(x) = sin πx 3. f(x) = (x2 - 1) cops 2x
Find the slopes of the tangent lines to the curve y = x3 - 3x at the points where x = - 2, -1.0, 1, 2.
Sketch the graph of y = 1/(x + 1) and then find the equation of the tangent line at (1, ½) (see Example 3).
Find the equation of the tangent line to y = 1/(x - 1) at (0, - 1).
Experiment suggests that a falling body will fall approximately 16t2 feet in t seconds. (a) How far will it fall between t = 0 and t = 1? (b) How far will it fall between t = 1 and t = 2? (c) What is
An object travels along a line so that its position s is s = t2 + 1 meters after t seconds. (a) What is its average velocity on the interval 2 ≤ t ≤ 3? (b) What is its average velocity on the
Suppose that an object moves along a coordinate line so that its directed distance from the origin after t seconds is √2t + 1 feet. (a) Find its instantaneous velocity at t = α, α > 0. (b) When
If a particle moves along a coordinate line so that its directed distance from the origin after t seconds it (-t2 + 4t) feet, when did the particle come to a momentary stop (i.e., when did its
A certain bacterial culture is growing so that it has a mass of ½ t2 + 1 grams after t hours. (a) How much did it grow during the interval 2 ≤ t ≤ 2.01? (b) What was its average growth rate
A business is prospering in such a way that its total (accumulated) profit after t years is 1000t2 dollars. (a) How much did the business make during the third year (between t = 2 and t = 3)? (b)
A wire of length 8 centimeters is such that the mass between its left end and a points x centimeters to the right is x3 grams (Figure 12).(a) What is the average density of the middle 2-centimeter
Suppose that the revenue R(n) in dollars from producing n computers is given by R(n) = 0.4n - 0.001n2. Find the instantaneous rates of change of revenue when n = 10 and n = 100. (The instantaneous
The rate of change of velocity with respect to time is called acceleration. Suppose that the velocity at time t of a particle is given by v(t) = 2t2. Find the instantaneous acceleration when t = 1
A city is hit by an Asian flu epidemic. Official estimates that t days after the beginning of the epidemic the number of persons sick with the flu is given by p(t) = 120t2 - 2t3, when 0 ≤ t ≤ 40.
The graph in Figure 13 shows the amount of water in a city water tank during one day when no water was pumped into the tank. What was the average rate of water usage during the day? How fast was
Passengers board an elevator at the ground floor (i.e., the 0th floor) and take it to the seventh floor. Which is 84 feet above the ground floor. The elevator's position s as a function of time t
Figure 15 shows the normal high temperature for St, Louis, Missouri, as a function of time (measured in days beginning January 1).(a) What is the approximate rate change in the normal high

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