- 4.1: If a rectangle has its base on the x-axis and two vertices on the c...
- 4.2: Show that | sin x 2 cos x | < s2 for all x.
- 4.3: Does the function fsxd e10| x22|2x 2 have an absolute maximum? If s...
- 4.4: Show that x 2 y 2 s4 2 x 2 ds4 2 y 2 d < 16 for all numbers x and y...
- 4.5: Show that the inflection points of the curve y ssin xdyx lie on the...
- 4.6: Find the point on the parabola y 1 2 x 2 at which the tangent line ...
- 4.7: If a, b, c, and d are constants such that lim xl0 ax 2 1 sin bx 1 s...
- 4.8: Evaluate lim xl` sx 1 2d 1yx 2 x 1yx sx 1 3d 1yx 2 x1yx
- 4.9: Find the highest and lowest points on the curve x 2 1 xy 1 y 2 12.
- 4.10: Sketch the set of all points sx, yd such that | x 1 y | < ex .
- 4.11: If Psa, a2 d is any point on the parabola y x 2 , except for the or...
- 4.12: For what values of c does the curve y cx 3 1 ex have inflection poi...
- 4.13: An isosceles triangle is circumscribed about the unit circle so tha...
- 4.14: Sketch the region in the plane consisting of all points sx, yd such...
- 4.15: The line y mx 1 b intersects the parabola y x 2 in points A and B. ...
- 4.16: ABCD is a square piece of paper with sides of length 1 m. A quarter...
- 4.17: For which positive numbers a does the curve y ax intersect the line...
- 4.18: For what value of a is the following equation true? lim xl` S x 1 a...
- 4.19: Let fsxd a1 sin x 1 a2 sin 2x 1 1 an sin nx, where a1, a2, . . . , ...
- 4.20: An arc PQ of a circle subtends a central angle as in the figure. Le...
- 4.21: The speeds of sound c1 in an upper layer and c2 in a lower layer of...
- 4.22: For what values of c is there a straight line that intersects the c...
- 4.23: One of the problems posed by the Marquis de lHospital in his calcul...
- 4.24: Given a sphere with radius r, find the height of a pyramid of minim...
- 4.25: Assume that a snowball melts so that its volume decreases at a rate...
- 4.26: A hemispherical bubble is placed on a spherical bubble of radius 1....
Solutions for Chapter 4: Calculus 8th Edition
Full solutions for Calculus | 8th Edition
Angle of elevation
The acute angle formed by the line of sight (upward) and the horizontal
Speed of rotation, typically measured in radians or revolutions per unit time
The notation PQ denoting the directed line segment with initial point P and terminal point Q.
De Moivre’s theorem
(r(cos ? + i sin ?))n = r n (cos n? + i sin n?)
Frequency (in statistics)
The number of individuals or observations with a certain characteristic.
Higher-degree polynomial function
A polynomial function whose degree is ? 3
A special case of a limit that does not exist.
The value of |r| at the point on the graph of a polar equation that has the maximum distance from the pole
Measure of center
A measure of the typical, middle, or average value for a data set
The multiplicity of a zero c of a polynomial ƒ(x) of degree n > 0 is the number of times the factor (x - c) (x - z 2) Á (x - z n)
See Numerical derivative of ƒ at x = a.
Partial fraction decomposition
See Partial fractions.
Principle of mathematical induction
A principle related to mathematical induction.
Product of functions
(ƒg)(x) = ƒ(x)g(x)
Solution of an equation or inequality
A value of the variable (or values of the variables) for which the equation or inequality is true
p = ƒ(x), where x represents production and p represents price
symmetric about the x-axis
A graph in which (x, -y) is on the graph whenever (x, y) is; or a graph in which (r, -?) or (-r, ?, -?) is on the graph whenever (r, ?) is
A matrix A = [aij] with the property aij = aji for all i and j
A function that maps real numbers to real numbers.
Upper bound for real zeros
A number d is an upper bound for the set of real zeros of ƒ if ƒ(x) ? 0 whenever x > d.