Questions and Answers of Calculus An Intuitive And Physical Approach

State in your own words what one means by the statement that y is a function of x.
A rocket is at rest. Then a force applied instantaneously at t = 0 gives the rocket an upward velocity of 100 ft/sec. Graph the velocity as a function of time from t = − ∞ to t = ∞. Assume that
Would you say that Bernoulli was able to solve the brachistochrone by relying entirely upon mathematics and concepts of mechanics such as velocity and acceleration?
Interpret geometrically f(xo + 4x) = f(xο - Δ.x) 2 Δx
Find the length of the longest side of the triangle whose vertices are (0, 0, 0), (3, 0, 0), and (3, 3, 0).
Indicate as best you can on a set of axes where the following points lie:(a) (2, 3, −5).(b) (−2,1,4).(c) (2, −3, −4).(d) (0,3,5).(e) (3,0,5).(f) (3, 5, 0).(g) (0,0,5).(h) (0, −5, 0).(i)
The triangle whose vertices are (0, 2, 0), (4, 2, 0), and (0, 2, 3) is a right triangle. Find the length of the hypotenuse.
What is the numerical distance of the point (x, y, z) from each of the three axes?
Two of the sides of the triangle whose vertices are (4, −2, 0), (4, 2, 0), and (0, 0, 5) are equal. What are their lengths?
When we speak of two points being symmetrically placed with respect to a plane, we mean that the line joining the two points is perpendicular to and bisected by the plane. What points are symmetric
Where are all the points for which the following hold?(a) x = 0.(b) y = 0.(c) z = 0.
Where are all the points for which the following hold?(a) x = 0 and y = 0.(b) x = 0 and z = 0.(c) y = 0 and z = 0.
Describe geometrically the following sets of points:(a) All points whose perpendicular distance from the xy-plane is 7.(b) All points whose perpendicular distance from the xz-plane is −5.
Describe geometrically the following sets of points:(a) All points whose perpendicular distance to the z-axis is 4.(b) All points whose perpendicular distances from the xz- and yz planes are equal.
If the slope of a curve at any point (x, y) is 2x + 3y, find the equation of the curve.
Solve equation (15) when R = 10 ohms, L is 2 henrys and E is 20 cos 51 volts.
Show that, when the roots of the characteristic equation of. y″ + 2αy′ βy = 0 are equal, and if m is the double root, then (A + Bx)emx is the general solution.
An electromotive force of 20 volts is applied to a circuit consisting of an inductance of 2 henrys and a resistance of 40 ohms. Find the current at any time t if it is 0 when t = 0.
(a) A bead is free to slide along a smooth straight horizontal wire. One end of the wire is kept at a fixed point which one can take as the pole or origin and the wire is rotated around the origin in
Find the current in a series circuit containing a resistance of 10 ohms, an inductance of 0.1 henry, and a voltage E(t) such that E(t) = 10 for 0 ≤ t ≤ 5 and E(t) = 0 for t > 5.
Solve the following differential equations by the method of separation of variables x√1+ y² dx -y V1 + x² dy = 0.
Solve the following differential equations by the method of separation of variablesy′ = 8xy + 3y.
Solve the following differential equations by the method of separation of variables.Solve y′ = (x + y)2 by letting v = x + y.
Is it permissible to replace the problem of findingby the problem of findingIf so, why? lim x→0 X² X X
What is the domain of a function? The range of a function?
Criticize the following reasoning:because x + 3 = 6 when x = 3. x²-9 lim x3 x3 lim x + 3 x→3 lim x + 3 = 6 X-3
Suppose that a rocket is shot up into the air with an initial velocity of 400 ft/sec and loses velocity at the rate of 32 ft/sec each second. After 6 seconds an explosive charge carried by the rocket
Draw a graph showing the domain and range of the function for positive and negativevalues of x. y =√x³ = x
Graph the following functions:(a)(b) y = x + √(1-x)(2 - x)
The attraction of a spherical shell (idealized as a spherical surface of radius R) of mass M per unit area on a unit particle inside is 0. The attraction of the shell on a unit particle outside is
Let [x] denote the integer (including 0) which is closest to x and let f(x) = |[x] − x|. Graph the function for x in the domain (0, 10).
Sketch the graph of y = f(x) where f(x) = x for −1 ≤ x ≤ 1 and f(x) is periodic and of period 2.
Sketch the graph of y = 1/(x2 −1).
Is it true for n = 0 that if y = xn then y′ = nxn −1?
Find the law of variation of the mass of a string suspended from two points at the same level and acted upon by gravity so that it hangs in the form of a semicircle. Take the semicircle to be the
Would you say that Bernoulli used an entirely mathematical argument to solve the brachistochrone problem?
A heavy chain is suspended at its two extremities and forms an arc of the parabola y = x2 /4p. Show that the weight per horizontal foot is constant.
Specifically, what did Bernoulli accomplish by introducing the motion of light?
Show that the projectile moving in the resisting medium attains its maximum height at a value of x closer to the starting point than it does when shot out at the same angle A and with the same
What is the essence of the argument that the cycloid requires least time?
(a) What is the terminal horizontal velocity, that is, the velocity as t becomes infinite, of the projectile motion discussed in the text?(b) What is the terminal vertical velocity?(c) Using the
For each of the following functions find the number or limit approached by the numerator, by the denominator, and by the entire function as h approaches 0:(a)(b)(c)(d)(e)(f) 3h2 h2
Sketch the graphs of the following functions by the method of plotting points:(a) y = 3x2.(b) y = √1 - x2(c) y = -√1 - x2(d) y = 1/1-3(e) f(x) = x3.(f) y = 1/x2.(g) y = (x2 - 9)/(x -
Sketch the graphs of the following functions by the method of plotting points:(a) y = 3x2.(b) y = √1 - x2(c) y = -√1 - x2(d) y = 1/1-3(e) f(x) = x3.(f) y = 1/x2.(g) y = (x2 - 9)/(x -
Suppose that in order to obtain the speed of an object at the end of the third second of its motion you calculated the average speed during the third second, then during the time interval from 2 to
Distinguish between the change in distance that results when an object moves for some interval of time and the rate of change of distance compared to time in that interval.
There is of course no obligation to use y and x as the symbols for the dependent and independent variables, respectively. We may use k and h. Sketch the following functions:(a) (b)(c)(d)(e) k
The amount A of money that accumulates in n years if one dollar is invested and if the interest is compounded annually at the fixed rate of i per cent per year is A = (1 + i)n. As the formula is
If f(x) = x2 −9x, calculate f(0), f(2), f(−1), and f(9).
Give the argument that convinces you that, as the values Δt approach 0, 16Δt also approaches 0.
Calculate the derivatives of the following functions at the specified values of the independent variable:(a) y = 16x2 at x = x1,(b) y = 16x2 at x = 4.(c) y = bx2 at x = 4.(d) s = 16t2 at t = 5.(e) A
Suppose that the average speeds of a moving object for smaller and smaller intervals of time after the third second prove to be 129, 128.5, 128.2, 128.05,…. What would you expect the speed at the
Describe in your own words the essence of the method of increments.
Since the functionis the same as the function y = x2 + x +1 except at x = 1 what is the limit of as x approaches 1? V (x³ - 1) x-1
Solve v = 32t for t. Is the resulting function single-valued?
If f(x) = − x2 −9x, calculate f(0), f(2), f( −2), f(9), and f( −9).
If s = 16t2, how much is Δs when t = 3 and Δt = 1? When t = 4 and Δt = 1? When t = 4 and Δt = 1/2?
Distinguish between average speed and instantaneous speed.
Suppose that the fall of an object is described by the formula s=16t2. Use the method of increments to calculate the following:(a) The instantaneous speed at the end of the third second of fall.(b)
Solve y = 5.3x2 for x. Is the resulting function single-valued?
Use the delta notation to calculate the instantaneous speed at the end of the fifth second of an object that falls according to the formula s = 16t2.
A man lays out a circular area of radius 100 feet. If he increases the radius by 10 feet, how much does he increase the area? When r1 = 100 and Δr = 10, how much is ΔA?
What mathematical concept is used to obtain instantaneous speed from average speed?
Suppose that the formula that relates the height above the ground and the time of travel of a ball thrown up into the air is s = 128t − 16t2.(a) How high is the ball when t = 3?(b) What is the
Write the formula for:(a) The area A of a circle in terms of the radius;(b) The area A of a circle in terms of the diameter.
If f(x) = x −1/x show that (a) f(−x) = −f(x),(b) f(1/x) = − f(x).
Iffind f(0), f(2), f( −2), and f(√7). x² - 9 f(x) = -x² + 7,
Use the delta notation to calculate the instantaneous speed at the instant t = t1 of an object that falls according to the formula s = 16t2.
Determine the derivative of y = x2 at x = x1 by the method of increments or any other legitimate method. Compare the results with that for y = ax2. Does the comparison suggest any general statement
If the distance s, in feet, that a body falls in t seconds is given by the formula s = 16t2, calculate the following:(a) The average speed of the body during the first 5 seconds of fall.(b) The
Find the limit as h approaches 0 of h/(√h + 4 -2).
Write the formula for the radius of a circle in terms of the area. Is the function single-valued? If not, which of the single-valued functions do you think would be more useful and why?
An object dropped near the surface of the moon falls to the surface in accordance with the formula s = 2.6t2. Use the delta notation to calculate the speed of the object at the end of the fourth
Find the rate of change of the area A of a square with respect to a side s at a given value s1 of the side. Is the result intuitively reasonable?
The area of a rectangle is given by the formula A = lw, where l and w are the length and width, respectively. Suppose that l is kept fixed. Find the rate of change of A with respect to w at a given
In the text we calculated the instantaneous speed at t = 4 of an object falling according to the formula s = 16t2 by first calculating average speed over intervals of time following t = 4; that is,
(a) Is the function (3h2 + h)/h identical with the function 3h + 1?(b) Can we use the latter function in place of the former to determine the number approached by the former as h approaches 0?
One arm of a right triangle is 3 units and the hypotenuse is x units. Write a formula for the length of the other arm.
If f(x) = x2 −7x, what is f(2x)? What is f(x + h)?
An object dropped near the surface of the sun falls to the surface in accordance with the formula s = 432t2. Use the delta notation to calculate the speed of the object at the end of the fifth second.
If at some instant during its motion an object has the speed of 30 mi/h, will the object travel 30 miles in the next hour?
Iffind f(3), f( −1), f(1/x). f(x)= 3 x
An automobile travels at 30 miles per hour. Write a formula which expresses the distance d traveled in feet as a function of the time t which represents the number of seconds of travel.
If f(x) = tan x, find f(0), f(π/4), and f(− π/4).
If y = f(x), what doesdenote in the delta notation? f(x)-f(xo) x xo
Consider the function s = t2. At Δt = 2 and for Δt = 0.1, Δs/Δt. is a good approximation to . How could you improve this approximation to s?
If y = f(x), what does f(x) − f(x0) denote in the delta notation?
A rectangle is required to have an area of 4 square feet but its dimensions may vary. If one side has length x, express the perimeter p of the rectangle as a function of x.
If y = f(x), what doesdenote? f(x)-f(xo) lin x-xo x-xo
If f(x) = x2 + 5 and g(x) = x3 −7, how much is f( −2) · g( −2).
Calculate the instantaneous speed of the following relations between distance and time at the instant indicated:(a) s = 4t2 at t = 3.(b) s = 1/4t2 at t = 3.(c) s = 3t2 at t = 0.(d) s = 5/2t2 at t = 2.
Iffind f(0), f(4), f(g2). f(x) = x² + 32 x + 4
Is a limit a variable or a constant?
If y = f(x), what does f′(x0) denote?
Is a limit an exact value or an approximate value?
In using the method of increments we encounter the function Δy/Δx. What are the independent and dependent variables in this function?
Let g(x) = x3. Show that g(− x) = − g(x).
Let g(x) = x4 + 2x2 + 1. Show that g(x) = g(− x).
We learn in trigonometry that sin x ≡ sin(π − x). Hence f(sin x) = f(sin[π − x]). Now let f(x) = x sin x. Then x sin x = (π − x) sin(π − x) or x = π − x. Hence π = 2x and since x is
Draw the line joining the two given points and calculate its slope:(a) (3, 4) and (5, 7).(b) (−1, 2) and (5, 7).(c) (3, 7) and (5, 2).(d) (−3, 7) and (5, 2).(e) (−3, 7) and (−5, −2).(f) (3,

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