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study help
mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Evaluate the indefinite integral, using substitution if necessary. (In x)² X -dx
Assume ƒ is differentiable. Which of the following statements does not follow from the MVT?(a) If ƒ has a secant line of slope 0, then ƒ has a tangent line of slope 0.(b) If ƒ (5) < ƒ(9), then ƒ'(c) > 0 for some c ∈ (5, 9).(c) If ƒ has a tangent line of slope 0, then ƒ has a secant line
Can a function with the real numbers as its domain that takes on only negative values have a positive derivative? If so, sketch an example.
For ƒ with derivative as in Figure 14:(a) Is ƒ(c) a local minimum or maximum?(b) Is ƒ a decreasing function? C X
Estimate using the Linear Approximation or linearization, and use a calculator to estimate the error.√101
State the Squeeze Theorem carefully. If h(x) ≤ g(x)/(x) when x is near a, except possibly at a, and lim f(x) = limh(x)=L, then lim g(x)=L. 3-0 1-N a adlo) h(x)
Estimate using the Linear Approximation or linearization, and use a calculator to estimate the error. 1 V4.1 - 2
Which is the correct conclusion, (a) or (b)?If ƒ is not continuous on [0, 1], then(a) ƒ has no extreme values on [0, 1].(b) ƒ might not have any extreme values on [0, 1].
Estimate g(1.2) − g(1) if g'(1) = 4.
Wire of length 12 m is divided into two pieces and each piece is bent into a square. How should this be done in order to minimize the sum of the areas of the two squares?(a) Express the sum of the areas of the squares in terms of the lengths x and y of the two pieces.(b) What is the constraint
What conclusion can you draw if ƒ'(c) = 0 and ƒ''(c) < 0?
Plot ƒ(x) = x3 + 4x2 − x − 4 and indicate on the graph where it appears that inflection points occur. Then find the inflection points using calculus.
Match each statement with a graph in Figure 14 that represents company profits as a function of time.(a) The outlook is great: The growth rate keeps increasing.(b) We’re losing money, but not as quickly as before.(c) We’re losing money, and it’s getting worse as time goes on.(d) We’re doing
Estimate using the Linear Approximation or linearization, and use a calculator to estimate the error.6251/4 − 6241/4
Estimate ƒ(2.1) if ƒ(2) = 1 and ƒ'(2) = 3.
A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 12 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary.
True or false? If ƒ(c) is a local min, then ƒ"(c) must be positive.
Which is the correct conclusion, (a) or (b)?If ƒ is continuous but has no critical points in [0, 1], then (a) ƒ has no min or max on [0, 1]. (b) Either ƒ(0) or ƒ(1) is the minimum value on [0, 1].
Complete the following sentence: The Linear Approximation shows that up to a small error, the change in output Δf is directly proportional to __________.
Plot ƒ(x) = x(x − 4)3 and indicate on the graph where it appears that inflection points occur. Then find the inflection points using calculus.
True or false? If ƒ"(c) = 0, then ƒ has an inflection point at x = c.
For each statement, indicate whether it is true or false. If false, correct the statement or explain why it is false.(a) If ƒ'(c) = 0, then ƒ (c) is either a local minimum or a local maximum.(b) If ƒ (c) is the absolute maximum of ƒ on an interval I, then ƒ'(c) = 0.(c) If ƒ is differentiable
Estimate using the Linear Approximation or linearization, and use a calculator to estimate the error.1/1.02
Which of the six standard trigonometric functions have infinitely many local minima and infinitely many local maxima but no absolute maximum and no absolute minimum over their whole domain?
Determine the intervals on which the function is concave up or down and find the points of inflection.y = x2 − 4x + 3
Determine the intervals on which the function is concave up or down and find the points of inflection.y = 10x3 − x5
Find the linearization at the point indicated.y = √x, a = 25
Determine the intervals on which the function is concave up or down and find the points of inflection.y = 5x2 + x4
Find the linearization at the point indicated.v(t) = 32t − 4t2, a = 2
Determine the intervals on which the function is concave up or down and find the points of inflection.y = θ − 2 sinθ, [0, 2π]
Find the linearization at the point indicated. A(r) = ਤੈr, a = 3
Determine the intervals on which the function is concave up or down and find the points of inflection.y = x7/5
Find the angle θ that maximizes the area of the isosceles triangle whose legs have length ℓ (Figure 17), using the fact the area is given by A = 1/2ℓ2 sin θ. l 0 l
Find the extreme values on the interval.ƒ(x) = x − x3/2, [0, 2]
Let ƒ(x) = x2 + 3x.Find a formula for LN for ƒ on [0, 2] and compute f(x) dx by taking the limit.
A quantity N(T) satisfies dN/dt = 2/t − 8/t2 for t ≥ 4 (t in days). At which time is N increasing most rapidly?
A projectile is released with an initial (vertical) velocity of 100 m/s. Use the formula v(t) = 100 − 9.8t for velocity to determine the distance traveled during the first 15 seconds.
A particle moves in a straight line with the given velocity (in meters per second). Find the displacement and distance traveled over the time interval, and draw a motion diagram like Figure 3 (with distance and time labels).v(t) = 12 − 4t, [0, 5]Figure 3 t=0 0 1=6 1=4 36 128 3 Distance
Calculate R5, M5, and L5 for ƒ(x) = (x2 + 1)−1 on the interval [0, 1].
Evaluate the integral using FTC I. -2 (10x² + 3x5) dx
Evaluate the integral using FTC I. Sm/₂ 1/2 8 | x3 dx
Estimate the total number of extinct families from t = 0 to the present, using MN with N = 544.
Evaluate the integral using FTC I. -3π/4 π/4 (2-csc²x) dx
Calculate the indefinite integral. 8 dx x
Show that a particle, located at the origin at t = 1 and moving along the x-axis with velocity v(t) = t−2, will never pass the point x = 2.
Evaluate the integral using FTC I. 1.57079 sec² tdt
Calculate the indefinite integral. fsir sin(4x - 9) dx
Show that a particle, located at the origin at t = 1 and moving along the x-axis with velocity v(t) = t−1/2, moves arbitrarily far from the origin after sufficient time has elapsed.
Write the integral as a sum of integrals without absolute values and evaluate. J-2 |x|dx
Solve the differential equation with the given initial condition. dy dx = 4x, y(1) = 4
In a free market economy, the demand curve is the graph of the function D that represents the demand for a specific product by the consumers in the economy at price q. It is not surprising that the curve is decreasing, as the demand drops as the price goes up. The supply curve is the graph of the
Write the integral as a sum of integrals without absolute values and evaluate. So Jo 13 - x|dx
Write the integral as a sum of integrals without absolute values and evaluate. 3 J-2 |x³|dx
Solve the differential equation with the given initial condition. dy = x dx -1/2, y(1) = 1
Write the integral as a sum of integrals without absolute values and evaluate. Jo |x² - 1|dx
Solve the differential equation with the given initial condition. dy dx = sec² x, y) = 2
Write the integral as a sum of integrals without absolute values and evaluate. S \cos x| dx
Solve the differential equation with the given initial condition. dy dt 1 + r sin 3t, y(л) = π
Write the integral as a sum of integrals without absolute values and evaluate. Su 10 x - 4x + 3dx
Solve the differential equation with the given initial condition. dy dt y (3)=0 = cos 3nt + sin 4лt, y
Evaluate the integral in terms of the constants. Š x³ dx
Find ƒ(t) if ƒ(t) = 1 − 2t, ƒ(0) = 2, and ƒ(0) = −1.
Evaluate the integral in terms of the constants. a Jb xdx
At time t = 0, a driver begins decelerating at a constant rate of −10 m/s2 and comes to a halt after traveling 500 m.Find the velocity at t = 0.
With y = ƒ(x) as in Figure 11, letFind formulas for A(x) and B(x) valid on [2, 4]. A(x) = So f(t) dt and = S₁₁ B(x) = f(t) dt
Evaluate the integral in terms of the constants. Si x dx
Use the given substitution to evaluate the integral. 0 dt √4t + 12 u = 4t + 12
Evaluate the integral in terms of the constants. -x (t³ + t) dt
With y = ƒ(x) as in Figure 11, let A(x) = So f(t) dt and = S₁₁ B(x) = f(t) dt
Use the given substitution to evaluate the integral. (x² + 1) dx (x² + 3x)4 S u = x³ + 3x
Evaluate the integral in terms of the constants. •S₁ Calculate f(x) dx, where f(x) = (12-x² for x ≤2 x³ for x > 2
Use the given substitution to evaluate the integral. /6 9/20 sin x cos¹ x dx, u= COS x
Evaluate the integral in terms of the constants. · 5²ª. Calculate f(x) dx, where f(x) = sin x -2 sinx for x ≤ T for x> ^
Use the given substitution to evaluate the integral. fse sec² (28) tan(20) de, u = tan(20)
Evaluate the integral in terms of the constants. L x x dx = 0 if n is an odd whole number. Explain graphically. Use FTC I to show that
Evaluate the integral in terms of the constants.Plot the function ƒ(x) = 3 sin x − x. Find the positive root of ƒ to three decimal places and use it to find the area under the graph of ƒ in the first quadrant.
Evaluate the integral in terms of the constants.Calculate F(4) given that F(1) = 3 and F(x) = x2. Express F(4) − F(1) as a definite integral.
Evaluate the integral in terms of the constants. With n > 0, does S x" dx get larger or smaller as n increases? Explain graphically.
Evaluate the integral in terms of the constants.Calculate G(16), where dG/dt = t−1/2 and G(9) = −5.
Evaluate the integral in terms of the constants. $S₁² ² x dx get larger or smaller as k increases? Explain graphically. With k > 1, does
Evaluate the integral in terms of the constants.Theorem 1 is stated with the assumption that a THEOREM 1 The Fundamental Theorem of Calculus, Part I Assume that a < b and that f is continuous on [a, b]. If F is an antiderivative of f on [a,b], then Sh a f(x) dx = F(b) - F(a) 1
Evaluate the integral in terms of the constants. Prove a famous result of Archimedes (generalizing Exercise 50): For r (a) Show that C has x-coordinate (r + s)/2.(b) Show that ABDE has area (s − r)3/4 by viewing it as a parallelogram of height s − r and base of length CF.(c) Show that ACE has
Evaluate the integral in terms of the constants.Show that the area of the shaded parabolic arch in Figure 6 is equal to four-thirds the area of the triangle shown. y a a+b 2 b ·X
Evaluate the integral in terms of the constants.(a) Apply the Comparison Theorem (Theorem 5 in Section 5.2) to the inequality sin x ≤ x (valid for x ≥ 0) to prove that(b) Apply it again to prove that(c) Verify these inequalities for x = 0.3. 1. ≤cos x ≤ 1 2
Evaluate the integral in terms of the constants.Verify these inequalities for x = 0.1. Why have we specified x ≥ 0 for sin x but not for cos x? 1 X - - x² 2 +3 6 cos x ≤ 1- - sin x ≤ x - 2 + 24 x3 x-5 + 6 120 (for x ≥ 0)
Calculate the next pair of inequalities for sin x and cos x by integrating the results of Exercise 53. Can you guess the general pattern?Data From Exercise 53Evaluate the integral in terms of the constants.Verify these inequalities for x = 0.1. Why have we specified x ≥ 0 for sin x but not for
Use FTC I to prove that if |ƒ'(x)| ≤ K for x ∈ [a, b], then |ƒ'(x) − ƒ(a)| ≤ K|x − a| for x ∈ [a, b].
Evaluate cos2 x dx as follows. First, show with a graph that I = J. Then, prove that I + J = 2π. = 5² I = -27 = 5.²² sin² x dx and J =
(a) Use Exercise 55 to prove that | sin a − sin b| ≤ |a − b| for all a, b.(b) Let (x) = sin(x + a) − sin x. Use part (a) to show that the graph of ƒ lies between the horizontal lines y = ±a.(c) Plot y = ƒ(x) and the lines y = ±a to verify (b) for a = 0.5 and a = 0.2.Data From Exercise
Prove 2 ≤ ∫21 2x dx ≤4 and 1/9 ≤ ∫21 3−x dx ≤ 1/3
Compute the area of the region in Figure 1(A) enclosed by y = 2 − x2 and y = −2. y -2 (A) 2 y=-2 y=2-x² -X
What is the area interpretation of ∫b a (ƒ(x) − g(x) dx if ƒ(x) ≥ g(x)?
Consider the region under the graph of the constant function ƒ(x) = h over the interval [0, r]. Give the height and the radius of the cylinder generated when the region is rotated about(a) The x-axis (b) The y-axis
Find the area of the region between y = 3x2 + 12 and y = 4x + 4 over [−3, 3] (Figure 12). -3 -1 50 25 y = 3x² + 12 + 2 y = 4x + 4 +X 3
Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume.ƒ(x) = x3, [0, 1]
What is the average value of ƒ on [0, 4] if the area between the graph of ƒ and the x-axis is equal to 12?
Which of the following is a solid of revolution?(a) Sphere (b) Pyramid (c) Cylinder (d) Cube
(a) Sketch the solid obtained by revolving the region under the graph of ƒ about the x-axis over the given interval, (b) Describe the cross section perpendicular to the x-axis located at x, and (c) Calculate the volume of the solid.ƒ(x) = x + 1, [0, 3]
Why is integration needed to compute the work performed in stretching a spring?
Let V be the volume of a right circular cone of height 10 whose base is a circle of radius 4 [Figure 16(A)].(a) Use similar triangles to find the area of a horizontal cross section at a height y.(b) Calculate V by integrating the cross-sectional area. (A) 10 0 R (B) h 0
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![THEOREM 1 The Fundamental Theorem of Calculus, Part I Assume that a < b and that f is continuous on [a, b].](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1700/4/7/4/421655b2e35c98ae1700474415615.jpg)
