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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
We know how to find the extreme values of a continuous function ƒ(x) by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers.
The function V(x) = x(10 - 2x)(16 - 2x), 0 < x < 5, models the volume of a box.a. Find the extreme values of V.b. Interpret any values found in part (a) in terms of the volume of the box.
The height of a body moving vertically is given bywith s in meters and t in seconds. Find the body’s maximum height. S 281² + vot + S0₂ g > 0,
Use the results of Exercise 81 to show that the function have inverses over their domains. Find a formula for dƒ -1/dx using Theorem 3.ƒ(x) = x5/3Exercise 81Show that increasing functions and decreasing functions are oneto- one. That is, show that for any x1 and x2 in I, x2 ≠ x1 implies ƒ(x2)
Use limits to find horizontal asymptotes for each function.a.b. y = x tan X
Consider the cubic function ƒ(x) = ax3 + bx2 + cx + d.a. Show that ƒ can have 0, 1, or 2 critical points. Give examples and graphs to support your argument.b. How many local extreme values can ƒ have?
The sum of two nonnegative numbers is 36. Find the numbers ifa. The difference of their square roots is to be as large as possible.b. The sum of their square roots is to be as large as possible.
Find ƒ′(0) for f(x) = Se-11x², 0, x = 0 x = 0.
Suppose that at any given time t (in seconds) the current i (in amperes) in an alternating current circuit is i = 2 cos t + 2 sin t. What is the peak current for this circuit (largest magnitude)?
An isosceles triangle has its vertex at the origin and its base parallel to the x-axis with the vertices above the axis on the curve y = 27 - x2. Find the largest area the triangle can have.
Then find the extreme values of the function on the interval and say where they occur.ƒ(x) = |x - 2| + |x + 3|, -5 ≤ x ≤ 5
A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft3, to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?
Then find the extreme values of the function on the interval and say where they occur.g(x) = |x - 1| - | x - 5 |, -2 ≤ x ≤ 7
Find the height and radius of the largest right circular cylinder that can be put in a sphere of radius 23.
Which of the following graphs shows the solution of the initial value problem(a)(b)(c)Give reasons for your answer. dy dx = 2x, y 4 when x = 1? =
Which of the following graphs shows the solution of the initial value problem(a)(b)(c)Give reasons for your answer. dy dx -x, y = 1 when x = -1?
The figure here shows two right circular cones, one upside down inside the other. The two bases are parallel, and the vertex of the smaller cone lies at the center of the larger cone’s base. What values of r and h will give the smaller cone the largest possible volume? 12' r 6' ---1 h
What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot. 6 0 y (8,6) 8 -Х
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
Then find the extreme values of the function on the interval and say where they occur.h(x) = |x + 2| - |x - 3|, -∞ < x < ∞
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
The accompanying figure shows a portion of the graph of a twice-differentiable function y = ƒ(x). At each of the five labeled points, classify y and y as positive, negative, or zero. 0 P y = f(x) Q R S T X
Then find the extreme values of the function on the interval and say where they occur.k(x) = |x + 1| + |x - 3 |, -∞ < x < ∞
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
Sketch a smooth connected curve y = ƒ(x) with f(-2) = 8, f(0) = 4, f(2)= 0, f'(x) > 0 for |x| > 2, f'(2) = f'(-2) = 0, f'(x) < 0 for x| < 2, f"(x) < 0 for x < 0, f"(x) > 0 for x > 0.
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
An open-top rectangular box is constructed from a 10-in.-by-16-in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume. Support your answers graphically.
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where ƒ′= 0. (In some exercises, you may have to use the
The sum of two nonnegative numbers is 20. Find the numbersa. If the product of one number and the square root of the other is to be as large as possible.b. If one number plus the square root of the other is to be as large as possible.
Sketch the graph of a twice-differentiable function y = ƒ(x) with the following properties. Label coordinates where possible. X x < 2 2 2 < x < 4 4 4 < x < 6 6 x > 6 y 1 4 7 y' < 0, 0, 0, 0, 0, 0, y' Derivatives y² II A A A II V y" y">0 y" y" > 0 y" = 0 y" < 0 y" < 0 y' < 0, y"
The graphs show the position s = ƒ(t) of an object moving up and down on a coordinate line. (a) When is the object moving away from the origin? Toward the origin? At approximately what times is the (b) Velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration
The accompanying graph shows the hypothetical cost c = ƒ(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing? Cost c = f(x) 20 40 60 80 100 120 Thousands of units produced x
Sketch the graph of a twice-differentiable function y = ƒ(x) that passes through the points (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2) and whose first two derivatives have the following sign patterns. y': y": + -2 -1 1 + 0 1 + 2
The accompanying graph shows the monthly revenue of the Widget Corporation for the past 12 years. During approximately what time intervals was the marginal revenue increasing? Decreasing? 0 y = r(t) 5 10 t
The graphs show the position s = ƒ(t) of an object moving up and down on a coordinate line. (a) When is the object moving away from the origin? Toward the origin? At approximately what times is the (b) Velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration
Suppose the derivative of the function y = ƒ(x) isAt what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection? y' = (x - 1)²(x - 2).
Suppose the derivative of the function y = ƒ(x) isAt what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection? - y' = (x - 1)²(x − 2)(x − 4).
Suppose that the second derivative of the function y = ƒ(x) isFor what x-values does the graph of ƒ have an inflection point? y" = x²(x - 2)²(x + 3).
Suppose that the second derivative of the function y = ƒ(x) isFor what x-values does the graph of ƒ have an inflection point? y" = (x + 1)(x - 2).
a. Suppose that the velocity of a body moving along the s-axis isi) Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.ii) Find the body’s displacement from t = 1 to t = 3 given that s = -2 when t = 0.iii) Now find the body’s displacement from
For x > 0, sketch a curve y = ƒ(x) that has ƒ(1) = 0 and ƒ′(x) = 1>x. Can anything be said about the concavity of such a curve? Give reasons for your answer.
Can anything be said about the graph of a function y = ƒ(x) that has a continuous second derivative that is never zero? Give reasons for your answer.
If b, c, and d are constants, for what value of b will the curve y = x3 + bx2 + cx + d have a point of inflection at x = 1? Give reasons for your answer.
The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line iswhere v0 and s0 are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem S 3 = 2/1² + vot + So, (1)
a. Find the coordinates of the vertex of the parabola y = ax2 + bx + c, a ≠ 0.b. When is the parabola concave up? Concave down? Give reasons for your answers.
For free fall near the surface of a planet where the acceleration due to gravity has a constant magnitude of g length-units/sec2, Equation (1) in Exercise 125 takes the formwhere s is the body’s height above the surface. The equation has a minus sign because the acceleration acts downward, in the
What can you say about the inflection points of a quadratic curve y = ax2 + bx + c, a ≠ 0? Give reasons for your answer.
Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of
When Apollo 15 astronaut David Scott dropped a hammer and a feather on the moon to demonstrate that in a vacuum all bodies fall with the same (constant) acceleration, he dropped them from about 4 ft above the ground. The television footage of the event shows the hammer and the feather falling more
What can you say about the inflection points of a cubic curve y = ax3 + bx2 + cx + d, a ≠ 0? Give reasons for your answer.
Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of
A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec2. How fast will the rocket be going 1 min later?
The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve y = e-x2. What dimensions give the rectangle its largest area, and what is that area? 0 y y = e²x² X
Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of
Find the values of constants a, b, and c so that the graph of y = ax3 + bx2 + cx has a local maximum at x = 3, local minimum at x = -1, and inflection point at (1, 11).
You are driving along a highway at a steady 60 mph (88 ft/sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 ft? To find out, carry out the following steps.1. Solve the initial value problem2. Find the value of t that makes
Find the values of constants a, b, and c so that the graph of y = (x2 + a)/(bx + c) has a local minimum at x = 3 and a local maximum at (-1, -2).
Find the absolute maximum and minimum values of each function on the given interval. y = x ln 2x - x, 1 e 2e' 2
A particle moves on a coordinate line with acceleration a = d2s/dt2 = 15√t - (3/√t), subject to the conditions that ds/dt = 4 and s = 0 when t = 1. Finda. The velocity v = ds/dt in terms of tb. The position s in terms of t.
The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve y = (ln x)/x2. What dimensions give the rectangle its largest area, and what is that area? 0.2 0.1 0 y y = In x X² X
Find the absolute maximum and minimum values of each function on the given interval. y = 10x(2 Inx), (0, e²]
Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of
Find the absolute maxima and minima of the functions and say where they are assumed. f(x) = ex/√x+1
Use a CAS to solve the initial value problem. Plot the solution curves. y' = + x, y(1) = -1 X
Use a CAS to solve the initial value problem. Plot the solution curves. y' 1 √4 - x²² y(0) = 2
Find the absolute maxima and minima of the functions and say where they are assumed. g(x) = √3-2x-x²
Graph ƒ(x) = 2x4 - 4x2 + 1 and its first two derivatives together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′ and ƒ′′.
Use a CAS to solve the initial value problem. Plot the solution curves. y" =+ √x, y(1) = 0, y'(1) = 0
Graph ƒ(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2π. Comment on the behavior of the graph of ƒ in relation to the signs and values of ƒ′′.
Use a CAS to solve the initial value problem. Plot the solution curves.y′ = cos2 x + sin x, y(π) = 1
A round underwater transmission cable consists of a core of copper wires surrounded by nonconducting insulation. If x denotes the ratio of the radius of the core to the thickness of the insulation, it is known that the speed of the transmission signal is given by the equation y = x2 ln (1/x). If
Write the sums without sigma notation. Then evaluate them. 2 k=1 6k k + 1
Write the sums without sigma notation. Then evaluate them. 3 k=1 k - 1 k
Write the sums without sigma notation. Then evaluate them. 4 Σ cos ka k=1
Which of the following express 1 + 2 + 4 + 8 + 16 + 32 in sigma notation?a.b.c. 6 Σ2-1 k=1
Write the sums without sigma notation. Then evaluate them. 3 k=1 (-1)*+¹ sin TT k
Graph ƒ(x) = x ln x. Does the function appear to have an absolute minimum value? Confirm your answer with calculus.
Which of the following express 1 - 2 + 4 - 8 + 16 - 32 in sigma notation?a.b.c. 6 Σ(-2)-1 k=1
Graph ƒ(x) = (sin x)sin x over [0, 3π]. Explain what you see.
Write the sums without sigma notation. Then evaluate them. 5 Σ sin ka k=1
Use finite approximations to estimate the area under the graph of the function usinga. A lower sum with two rectangles of equal width.b. A lower sum with four rectangles of equal width.c. An upper sum with two rectangles of equal width.d. An upper sum with four rectangles of equal width.ƒ(x) = x2
Which formula is not equivalent to the other two?a.b.c. 4 k=2 (-1)k-1 k - 1
Which formula is not equivalent to the other two?a.b.c. 4 Σ( – 1)2 k - k=1
Write the sums without sigma notation. Then evaluate them. 4 Σ(-1)* cos kπ k=1
Use finite approximations to estimate the area under the graph of the function usinga. A lower sum with two rectangles of equal width.b. A lower sum with four rectangles of equal width.c. An upper sum with two rectangles of equal width.d. An upper sum with four rectangles of equal width.ƒ(x) = x3
Use finite approximations to estimate the area under the graph of the function usinga. A lower sum with two rectangles of equal width.b. A lower sum with four rectangles of equal width.c. An upper sum with two rectangles of equal width.d. An upper sum with four rectangles of equal width.ƒ(x) = 1/x
Use finite approximations to estimate the area under the graph of the function usinga. A lower sum with two rectangles of equal width.b. A lower sum with four rectangles of equal width.c. An upper sum with two rectangles of equal width.d. An upper sum with four rectangles of equal width.ƒ(x) = 4 -
You and a companion are about to drive a twisty stretch of dirt road in a car whose speedometer works but whose odometer (mileage counter) is broken. To find out how long this particular stretch of road is, you record the car’s velocity at 10-sec intervals, with the results shown in the
The accompanying table shows the velocity of a model train engine moving along a track for 10 sec. Estimate the distance traveled by the engine using 10 subintervals of length 1 witha. Left-endpoint values.b. Right-endpoint values.
The accompanying table gives data for the velocity of a vintage sports car accelerating from 0 to 142 mi / h in 36 sec (10 thousandths of an hour).a. Use rectangles to estimate how far the car traveled during the 36 sec it took to reach 142 mi / h.b. Roughly how many seconds did it take the car to
Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle’s base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles.ƒ(x) = x2 between x = 0 and x = 1.
You are sitting on the bank of a tidal river watching the incoming tide carry a bottle upstream. You record the velocity of the flow every 5 minutes for an hour, with the results shown in the accompanying table. About how far upstream did the bottle travel during that hour? Find an estimate using
Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle’s base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles.ƒ(x) = x3 between x = 0 and x = 1.
An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in ft/sec2 and recorded every second after the drop for 5 sec, as shown:a. Find
Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. 1 1 1 + 2 4 8 + + 1 16
Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle’s base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles.ƒ(x) = 1/x between x = 1 and x = 5.
Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle’s base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles.ƒ(x) = 4 - x2 between x = -2 and x = 2.
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