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mathematics
calculus

Questions and Answers of Calculus

The point (2, 3), (6, 3), (6, - 1), and (2, -1) are corners of a square. Find the equations of the inscribed and circumscribed circles.
Show that the triangle whose vertices are (5, 3), (-2, 4), and (10, 8) is isosceles.
A belt fits tightly around the two circles, with equations (x - 1)2 + (y + 2)2 = 16 and (x + 9)2 + (y - 10)2 = 16. How long is this belt?
Show that the midpoint of the hypotenuse of any right triangle is equidistant from the three vertices.
Find the equation of the circle circumscribed about the right triangle whose vertices are (0, 0) (8, 0), and (0, 6).
Show that the two circle x2 + y2 - 4x - 2y - 11 = 0 and x2 + y2 + 20x - 12y + 72 = 0 do not intersect.
What relationship between a, b, and c must hold if x2 + ax + y2 + by + c = 0 is the equation of a circle?
The ceiling of an attic makes an angle of 30° with the floor. A pipe of radius 2 inches is placed along the edge of the attic in such a way that one side of the pipe touches the ceiling and
A circle of radius I? is placed in the first quadrant as shown in Figure 18. What is the radius r of the largest circle that can be placed between the original circle and the origin?
Construct a geometric proof using Figure 15 that shows two lines are perpendicular if and only if their slopes are negative reciprocals of one another.
Show that the set of points that are twice as far from (3, 4) as from (1, 1) form a circle. Find its center and radius.
The Pythagorean Theorem says that the areas A, B, and C of the squares in Figure 19 satisfy A + B = C. Show that semi-circles and equilateral triangles satisfy the same relation and then guess what a
Show that the triangle whose vertices are (2, -4), (4, 0), and (8, - 2) is a right triangle.
Consider a circle C and a point P exterior to the circle, Let line sement PT be tangent to C at T, and let the line through P and the center of C intersect C at M and N. Show that (PM) (PN) = (PT)2.
A belt fits around the three circles x2 + y2 = 4, (x - 8)2 + y2 = 4, and (x - 6)2 + (y - 8)2 = 4, as shown in Figure 20. Find the length of this belt.
Study Problems 50 and 61. Consider a set of nonintersecting circles of radius r with centers at the vertices of a convex n-sided polygon having sides of lengths d1, d2, . . . , dn. How long is the
Use this result to find the distance from the given point to the given line. 1. (-3, 2); 3x + 4y = 6 2. (4, - 1); 2x - 2y + 4 = 0 3. (-2, - 1); 5y = 12x = 1 4. (3, - 1); y = 2x - 5
In problem 67 and 68 find the (perpendicular) distance between the given parallel lines. 1. 2x + 4y = 7, 2x + 4y = 5 2. 7x - 5y = 6, 7x - 5y = - 1
Find the equation for the line that bisects the line segment from (-2, 3) to (1, - 2) and is at right angles to this line segment.
The center of the circumscribed circle of a triangle lies on the perpendicular bisectors of the sides. Use this fact to find the center of the circle that circumscribes the triangle with vertices (0,
Find the radius of the circle that is inscribed in a triangle with sides of lengths 3, 4, and 5 (see Figure 21).
Suppose that (a, b) is on the circle x2 + y2 = r2. Show that the line ax + by = r2 is tangent to the circle at (a, b).
Find the equations of the two tangent lines to the circle x2 + y2 = 36 that go through (12, 0).
Express the perpendicular distance between the parallel lines y = mx + b and y = mx + B in terms of m, b, and B.
Show that the line through the midpoints of two sides of a triangle is parallel to the third side.
Show that the line segments joining the midpoints of adjacent sides of any quadrilateral (four-sided polygon) form a parallelogram.
Find the point on the x-axis that equidistant from (3, 1) and (6, 4).
In problem 1-3 plot the graph of each equation. Begin by checking for symmetries and be sure to find all x- and y-intercepts. 1. y = - x2 + 1 2. x = - y2 + 1 3. x2 + y = 0
In problem 1-3 plot the graph of both equations on the same coordinate plane, Find and label the points of intersection of the two graphs (see Example 4). 1. y = -x + 1 Y = (x + 1)2 2. y = 2x + 3 y =
Choose the equation that corresponds to each graph in Figure 8.(a) y = ax2, with a > 0(b) y = ax3 + bx2 + cx + d, with a > 0(c) y = ax3 + bx3 + cx + d, with a (d) y = ax3, with a > 0
Find the distance between the points on the circle x2 + y2 = 13 with the x-coordinates -2 and 2. How many such distances are there?
Find the distance between the points on the circle x2 + 2x + y2 - 2y = 20 with the x-coordinates - 2 and 2. How many such distances are there?
For f(x) = 1 - x2, find each value. (a) f(1) (b) f(-2) (c) f(0) (d) f(k) (e) f(-5) (f) f(1/4) (g) f(1 + h) (h) f(1 + h) - f(1) (i) f(2 + h) - f(2)
For F(t) = 4t3, find and simplify [F(a + h) - F(a)] / h.
For g(u) = 3 / (u - 2), find and simplify [g(x + h) - g(x)] / h.
For G(t) = t / (t + 4), find and simplify [G(a + h) - G(a)] / h.
Find the natural domain for each of the following. (a) F(z) = √2z + 3 (b) g(v) = 1/(4v - 1) (c) ψ(x) = √x2 - 9 (d) H(y) = - √625 - y4
Find the natural domain in each case. (a) f(x) = 4 - x2 (b) G(y) = √(y + 1)-1 (c) ϕ(u) = |2u + 3| (d) F(t) = t2/3 - 4
In problem 1-3 specify whether the given function is even. Odd, or neither and then sketch its graph. 1. f(x) = - 4 2. f(x) = 3x 3. F(x) = 2x + 1
For F(x) = x2 + 3x, find each value. (a) F(1) (b) F(√2) (c) F(1/4) (d) F(1 + h) (e) F(1 + h) - F(1) (f) F(2 + h) - F(2)
For G(y) = 1/(y - 1), find each value. (a) G(0) (b) F(0.999) (c) G(1.01) (d) G(y2) (e) G(-x) (f) G(1/x2)
A plant has the capacity to produce from 0 to 100 computers per day. The daily overhead for the plant is $5000, and the direct cost (labor and materials) of producing one computer is $805. Write a
It costs the ABC Company 400 + 5√x(x - 4) dollars to make x toy stoves that sell for $6 each. (a) Find a formula for P(x), the total profit in making x stoves. (b) Evaluate P(200) and P(1000). (c)
Find the formula for the amount E(x) by which a number x exceeds its square. Plot a graph of E(x) for 0 ≤ x ≤ 1. Use the graph to estimate the positive number less than or equal to 1 that exceeds
Let p denote the perimeter of an equilateral triangle. Find a formula for A(p), the area of such a triangle.
A right triangle has a fixed hypotenuse of length h and one leg that has length x. Find a formula for the length L(x) of the other leg.
A right circular cylinder of radius r is inscribed in a sphere of radius 2r. Find a formula for V(r), the volume of the cylinder, in terms of r.
A 1-mile track has parallel sides and equal semicircular ends. Find a formula for the area enclosed by the track, A(d), in terms of the diameter d of the semicircles. What is the natural domain for
For φ (u) = u + u2 / √u, find each value. (ϕ is the uppercase Greek letter phi.) (a) ϕ(1) (b) ϕ(-t) (c) ϕ(1/2) (d) ϕ(u + 1) (e) ϕ(x2) (f) ϕ(x2 + x)
Let A(c) denote the area of the region bounded from above by the line y = x + 1, from the left by the y-axis, from below by the x-axis, and from the right by the line x = c. Such a function is called
Let B(c) denote the area of the region bounded from above by the graph of the curve y = x(1 - x), from below by the x-axis, and from the right by the line x = c. The domain of B is the interval [0,
Which of the following functions satisfies f(x + y) = f(x) + f(y) for all real numbers x and y? (a) f(t) = 2t (b) f(1) = t2 (c) f(t) = 2t + 1 (d) f(t) = - 3t
Let f(x + y) = f(x) + f(y) for all x and y. Prove that there is a number in such that f (t) = nit for all rational numbers t.
A baseball diamond is a square with sides of 90 feet. A player, after hitting a home run, loped around the diamond at 10 feet per second. Let s represent the player's distance from home plate after t
Let f(x) = (x3 + 3x - 5) / (x2 + 4). a. Evaluate f(1.38) and f(4.12). (b) Construct a table of values for this function corresponding to x = - 4, - 3 tan) / cos x. To use technology effectively, you
Follow the instruction in problem 45 for f(x) = (sin2 x - 3 tan x) / cos x. Let f(x) = (x3 + 3x - 5) / (x2 + 4). a. Evaluate f(1.38) and f(4.12). (b) Construct a table of values for this function
Draw the graph of f(x) = x3 - 5x2 + x + 8 on the domain [-2, 5]. (a) Determine the range of f. (b) Where on this domain f(x) ≥ 0? To use technology effectively, you need to discover its
Superimpose the graph of g(x) = 2x2 - 8x - 1 with domain [-2, 5] on the graph of f(x) of problem 47. (a) Estimate the x-values where f(x) = g(x). (b) Where on [-2, 5] is f(x) ≥ g(x)? (c) Estimate
Graph f(x) = (3x - 4) / (x2 + x - 6) on the domain [- 6, 6]. (a) Domain the x- and y-intercepts. (b) Determine the range of f for the given domain. (c) Determine the vertical asymptotes of the
For F(x) = 1 / √x - 3 Find each value (a) f(0.25) (b) f(π) (c) f(3 + √2)
Follow the directions in problem 49 for the function g(x) = (3x2 - 4) / (x2 + x - 6) Graph f(x) = (3x - 4) / (x2 + x - 6) on the domain [- 6, 6]. (a) Domain the x- and y-intercepts. (b) Determine the
For f(x) = √x2 + 9 / (x - √3), find each value. (a) f(0.25) (b) f(12.26) (c) f(√3)
Which of the following determine a function f with formula y = f (x)? For those that do, find f(x). (a) x2 + y2 = 1 (b) xy + y + x = 1, x ≠ - 1 (c) x = √2y + 1. (d) x = y/y + 1
For f(x) = x + 3 and g(x) = x2, find each value (if possible). (a) (f + g) (2) (b) (f ͦ g) (0) (c) (g / f) (3) (d) (f ͦ g) (1) (e) (g ͦ f) (1) (f) (g ͦ f) (- 8)
Write p(x) = 1 √x2 + 1 as a composite of four functions.
Sketch the graph of f(x) = √x - 2 - 3 by first sketching g(x) = √x and then translating. (See Example 4.)
Sketch the graph of g(x) = |x + 3| - 4 by first sketching h(x) = |x| and then translating.
Sketch the graph of f(x) = (x - 2)2 - 4 using translations.
Sketch the graph of f(x) = (x - 2)2 - 4 using translations. Discuss.
Sketch the graphs of f(x) = (x - 3) / 2 and g(x) = √x using the same coordinate axes. Then sketch f + g by adding y-coordinates.
For f(x) = x2 + x and g(x) = 2 / (x + 3), find each value. (a) (f - g) (2) (b) (f / g)(1) (c) g2(3) (d) (f ͦ g) (1) (e) (g ͦ f) (1) (f) (g ͦ f) (3)
Follow the directions of Problem 19 for f(x) = x and g(x) = |x|. In problem 19 Sketch the graphs of f(x) = (x - 3) / 2 and g(x) = √x using the same coordinate axes. Then sketch f + g by adding
Sketch the graph of F(t) = |t| - t / t.
Sketch the graph of G(t) = t - [t].
State whether each of the following is an odd function, an even function, or neither. Prove your statements. (a) The sum of two even functions (b) The sum of two odd functions (c) The product of two
Let F be any function whose domain contains - x when-ever it contains x. Prove each of the following. (a) F(x) - F (- x) is an odd function. (b) F(x) + F(- x) is an even function. (c) F can always be
Is every polynomial of even degree an even function? Is every polynomial of odd degree an odd function? Explain.
Classify each of the following as a PF (polynomial function), RF (rational function but not a polynomial function), or neither. (a) f(x) = 3x1/2 + 1 (b) f(x) = 3 (c) f(x) = 3x2 + 2x-1 (d) f(x) = πx3
The relationship between the unit price P (in cents) for a certain product and the demand D (in thousands of units) appears to satisfy P = √29 - 3D + D2 On the other hand, the demand has risen
Starting at noon, airplane A flies due north at 400 miles per hour. Starting 1 hour later, airplane B flies due east at 300 miles per hour. Neglecting the curvature of the Earth and assuming that
For φ(u) = u3 + 1 and ψ(v) = 1/v, find each value. (a) (φ + ψ) (t) (b) φ ͦ ψ) (r) (c) (ψ ͦ φ) (r) (d) φ3(z) (e) (φ - ψ) (5t) (f) ((φ - ψ) ͦ ψ) (t)
Let f(x) = ax + b / cx - a. Show that f(f(x)) = x, provided a2 + bc ≠ 0 and x ≠ a/c.
Let f(x) = x - 3 / x + 1. Show that f(f(f(x))) = x, provided x ≠ ± 1.
Let f(x) = x / x - 1. Find and simplify each value. (a) f(1/x) (b) f(f(x)) (c) f(1/f(x))
Let f(x) = x / √x - 1. Find and simplify (a) f(1 / x) (b) f(f(x))
Prove that the operation of composition of functions is associative; that is, f1 ͦ (f2 ͦ f3) = (f1 ͦ f2) ͦ f3.
Let f(x) = x2 - 3x. Using the same axes, draw the graphs of y = f(x), y = f(x - 0.5) - 0.6, and y = f(1.5x), all on the domain [-2, 5].
Let f(x) = |x3|. Using the same axes, draw the graphs of y = f(x), y = f(3x), and y = f(3(x - 0.8)), all on the domain [-3, 3].
Let f(x) = 2√x - 2x + 0.25 x2. Using the same axes, draw the graphs of y = f(x), y = f(1.5 x), and y = f(x - 1) + 0.5, all on the domain [0, 5].
If f(x) = √x2 - 1 and g(x) = 2/x, find formulas for the following and state their domains. (a) (f ͦ g) (x) (b) f4(x) + g4 (x) (c) (f ͦ g) (x) (d) (g ͦ f) (x)
Let f(x) = 1 / (x2 + 1). Using the same axes, draw the graph of y = f(X), y = f(2x), and y = f(x - 2) + 0.6, all on the domain [-4, 4].
Your computer algebra system (CAS) may allow the use of parameters in defining functions. In each case, draw the graph of y = f(x) for the specified values of the parameter k, using the same axes and
Using the same axes, draw the graph of f(x) = |k (x - c)|n for following choices of parameters.
If f(s) = √s2 - 4 and g(w) = |1 + w|, find formulas for (f ͦ g) (x) and (g ͦ f) (x).
If g(x) = x2 + 1, find formulas for g3(x) and (g ͦ g ͦ g) (x).
Convert the following degree measures to radians (leave π in your answer). (a) 30° (b) 45° (c) - 60° (d) 240° (e) -370° (f) 10°
Evaluate without using a calculator. (a) Tan π/3 (b) Sec π/3 (c) Cot π/3 (d) Csc π/4 (e) Tan (- π/6) (f) Cos (- π/3)
Verify that the following are identities (see Example 6). (a) (1 + sin z) (1 - sin z) = 1/ sec2 z (b) (sec t - 1) (sec t + 1) = tan2 t (c) Sec t - sin t tan t = cos t (d) Sec2 t - 1/sec2 t = sin2 t
Verify that the following are identities (see Example 6). (a) Sin2 v + 1/ sec2 v = 1 (b) Cos 3t = 4 cos3 t - 3 cos t (c) Sin 4x = 8 sin x cos3 x - 4 sin x cos x (d) (1 + cos θ) (1 - cos θ) = sin2 θ
Verify the following are identities. (a) sin u/csc u + cos u / sec u = 1 (b) 1 - cos2 x) (1 + cot2 x) = 1 (c) sin t (csc t - sin t) = cos2 t (d) 1 - csc2 t/ csc2 t = - 1 / sec2 t

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