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mathematics
calculus an intuitive and physical approach

Questions and Answers of Calculus An Intuitive And Physical Approach

Show that the line through (1,2) and (7, 3) is parallel to the line through (−4, −3) and (8, −1).
Find the distance in the following cases:(a) From the point (2, 1) to the line 4x − 3y + 15 = 0(b) From the point (3, 2) to the line 4x − y + 2 = 0.(c) From the point (−1, 3) to the line 4x +
Suppose that an object starts from rest and slides 200 feet down a hill whose inclination is 30°. How long does it take to slide this distance and what velocity does the object acquire?
Show that the line joining (1, 4) and (3, 7) is perpendicular to the line through (1, −4) and (4, −6).
Find the angle between the line through (1, 3) and (5, 7) and the line through (1, 3) and (−2, 7).
Suppose that an object starts from rest and slides down a hill whose inclination is 30°. The object slides from a height of 100 feet above the horizontal ground to the ground. How long does it take
Show that in finding the distance between two points it does not matter which point we denote by P1 and which by P2.
What is the inclination of a line which is perpendicular to the line through the points (2, 1) and (5, 4)?
Show that (−1, −4), (5, 4), and (−6, 6) are the vertices A, B, and C respectively of an isosceles triangle.
What is the inclination of a line which is perpendicular to a line whose slope is 2?
Show that the triangle whose vertices are (0, 5), (−3, 0), and (3, 0) is or is not equilateral.
What is the inclination of any line perpendicular to the line through the points (2, 1) and (−5, 7)?
Suppose that an object slides down a hill whose inclination is A and that at the instant it starts to slide it has a velocity of v0 ft/sec. Derive the formula for the distance the object slides in
With reference to the derivation of the equation of the cable, (a) Does the horizontal component of the tension T vary from point to point on the cable? (b) Does the vertical component of
By |x| we mean the positive value of x, whether the value of x in question be negative or positive. Thus | −3| = 3 and |3| = 3. Graph y = |x| for negative and positive x. What can you say about the
Assuming that w, the load per horizontal foot, is given, calculate the tension T at any point on the parabola y = (w/2T0)x2.Square (15), square (16), and add. Then use the equation of the parabola to
Does y = |x| have a derivative at x = 0?
Graph the function y = (x + |x|). At what values of x does the derivative exist and what is the value of the derivative when it does exist?
We have required in our derivation of the equation of the cable that the weight of the roadway per horizontal foot be constant. Can the roadway itself be curved?
Suppose that the load on the cable treated in the text were not the constant weight w per horizontal foot but that the load varied with the horizontal distance from the midpoint of the bridge in
For each of the following function find the derived function:(a) y = x6(b) y = x(c) y = x5(d) y = 4x5(e) y = x10(f) y = 7(g) y = x8(h) y = (1 / 2)x7
Associated with each point P of a curve (Fig. 6-7) is its abscissa x and the angle of inclination A of the tangent at P. Corresponding to each value of x in the interval OC there is a unique value of
Show that the rate of change of the volume of a cube with respect to the edge is three times the area of any face.
The statement that y approaches ∞ describes (a) The number that y approaches, (b) A state of mind, (c) The fact that the values of y surpass any number however large, (d) The
Sketch the graphs of the following functions. Use all the information that you can obtain from studying the first and second derived functions of the functions.
Suppose that the (total) cost of producing x units of a product is C(x) = x3 −6x2 + 15x. Find the number of units for which the average cost A(x) is a minimum. Verify that for this number of units
Suppose that the (total) cost of producing x units is C(x) = x3 −15x2 + 76x + 10 and that the demand function is p(x) = 55 −3x. Find the number of units for which the profit will be a maximum.
Suppose that the cost function of a commodity is C = 3x and the demand function is p = 10 −3x. The government imposes a sales tax of 25% (to be paid by the purchaser, or course). Find the maximum
Prove generally that when the profit is a maximum, the marginal revenue dR / dx equals the marginal cost.
Suppose that the demand function of some article is p(x) = 75 −2x and the cost function is C(x) = 350 + 12x + x2/4. Find the number of units and price at which the total profit is a maximum. What
Find the points of inflection of y = x4 −8x3 + 64x + 8.
Ifdu, what is d2g / dx2? 8(x)=√√u² + 2
A manufacturer sells x units of a product when the price p(x) per unit is 100 −0.10x dollars. The cost of x units is C(x) = 1000 + 50x dollars. How many should he sell to maximize his profit?
The manufacturing cost of an article involves a fixed overhead of $100 per day, 50 cents for material, and x2/100 dollars per day for labor and machinery to produce x articles. How many articles
The total cost of producing x units of an article is C(x) = a + c√x. What number of units will minimize average cost?
The amplitude of a body undergoing simple harmonic motion is a and its period is 1/5 second. Find its maximum velocity.
Evaluate d/dx ∫ab x3 dx when a and b are constants.
Sketch on the same axes y = sin x and(a) y = sin 3x.(b) y = 3 sin x.(c) y = 3 sin 2x.(d) y = 2 sin 4x.(e) y = sin(x +π/2).(f) y = sin x + π/2(g) y = sin (x − 1).(h) y = −2 sin 3x.
Express the area under the curve of y = 9 − x2 between x = 0 and x = 1 as a definite integral and then calculate it.
Ifwhat is dg / dx? 8(x)=f(u)du,
Sketch y = x + sin x.
Suppose a function y = f(x) is known to us only through the following table: x 0 2 3 1.72 1.72 1.60 1.44 1.24 4 5 6 7 1.06 0.92 0.80 0.70 0.92 0.80 0.70 8 0.63 0.63 9 10 0.56 0.50
Express the area between y = x3 + 9, y = x2, x = 1, and x = 5 as a definite integral and then evaluate it.
Sketch y = x sin x.
Find the area between the curves y = 3x2 and y = 5x2 and between the ordinates at x = 2 and x = 4.
Sketch y = 3 sin 2(x − 1) + 4.
Find the area between the curve y = 1 / x2 and the x-axis and between x = 1 and x = 3.
Criticize the following argument which “proves” that every triangle is isosceles. Consider triangle ABC (Fig. 9-17) and let AD be the altitude from A to BC. Now let PQ be any parallel to BC and
Prove with the aid of the fundamental theorem that fºcy dx= cf" y dx.
Sketch y = x2 + sin x.
Find the area between the curve y = x2 and the line y = 2x.
Find the area between the curves y = x2 and y = √5
Find the area between the curves y = 9 − x2 and y = x2.
Find the area between the curve y = x2 and the straight line y = 8x − 4.
Given y = sin x, find d2y / dx2.
Show by using a counterexample thatwhere u and v are functions of x. uv dx an "S+ xp a "S - xp n "s
Find the area between the parabolas y = 2x2 + 1 and y = x2 + 5.
Given y = cos x, find d2y / dx2.
Find the area between y2 = 16x and y2 = x3.
A swinging pendulum is 4 feet long and is rotating at the rate of 18° / sec when it makes an angle of 30° with the vertical. How fast is the end of the pendulum rising or falling at that moment?
Given log 7 = 0.8451, find the following:(a) log 70.(b) log 700.(c) log 0.7.(d) log 0.07.(e) log 0.007.
On the same set of axes sketch each of the following pairs of curves:(a) y = ex and y = 3ex.(b) y = ex and y = e−x.(c) y = ex and y = e(x+2).(d) y = ex and y = ex + 2.(e) y = ex and y = x + ex.(f)
Given log 7 = 0.8451 and log 9 = 0.9542, find the following:(a) log 63.(b) log 73.(c) log(90/7).(d) log 92. 7.(e) log 21.

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