Find an equation of the line that satisfies the given conditions. Slope 2/5, y-intercept 4
> Prove the identity. sin θ cot θ = cos θ
> Prove the identity. sin (π - x) = sin x
> Prove each equation. (a). Equation 10a (b). Equation 10b
> Find, correct to five decimal places, the length of the side labeled x. 40° 25 cm
> Find, correct to five decimal places, the length of the side labeled x. 10 cm 35°
> Find the remaining trigonometric ratios. tan a = 2, 0 < α < π/2
> Find the remaining trigonometric ratios. Sin θ = 3/5, 0 < θ < π/2
> Convert from degrees to radians. (a). 2100 (b). 90
> Sketch the graph of the equation. xy = 0
> Sketch the graph of the equation. y = -2
> Sketch the graph of the equation. x = 3
> Find the area inside the larger loop and outside the smaller loop of the limaçon r = 1/2 + cos θ.
> (a). Show that the points A (-1, 3), B (3, 11), and C (5, 15) are collinear (lie on the same line) by showing that |AB| + |BC| = |AC|. (b). Use slopes to show that A, B, and C are collinear.
> Show that the points (-2, 9), (4, 6), (1, 0), and (-5, 3) are the vertices of a square.
> Find an equation of the perpendicular bisector of the line segment joining the points A (1, 4) and B (7, -2).
> Find the midpoint of the line segment joining the points (1, 3) and (7, 15).
> Show that the midpoint of the line segment from P1 (x1, y1) to P2 (x2, y2) is (x1 + x2/2, y1 + y2/2).
> Show that the lines 3x – 5y + 19 = 0 and 10x + 6y – 50 = 0 are perpendicular and find their point of intersection.
> Show that the lines 2x – y = 4 and 6x – 2y = 10 are not parallel and find their point of intersection.
> Show that the equation represents a circle and find the center and radius. x² + y² + 6y + 2 = 0
> Find the slope of the line through P and Q. P (-1, -4), Q (6, 0)
> Show that the equation represents a circle and find the center and radius. x? + y? – 4x + 10y + 13 = 0
> Find the area of the region that lies inside both curves. r = 3 + 2 cos e, r= 3 + 2 sin e
> Find an equation of a circle that satisfies the given conditions. Center (-1, 5), passes through (-4, -6)
> Find an equation of a circle that satisfies the given conditions. Center (3, -1), radius 5
> Sketch the region in the xy-plane. {(x, y) | -x < y < 1/2 (x + 3)}
> Sketch the region in the xy-plane. {(x, y) |1 + x < y < 1 – 2x}
> Sketch the region in the xy-plane. {(x, y) | y > 2x – 1}
> Sketch the region in the xy-plane. {(x, y) |0 < y < 4 and x < 2}
> Sketch the region in the xy-plane. {(x. y) ||x| < 3 and |y| < 2}
> Sketch the region in the xy-plane. {(x, y) ||x| < 2
> Sketch the region in the xy-plane. {(x, y) | x > 1 and y < 3}
> Find the slope of the line through P and Q. P (-3, 3), Q (-1, -6)
> (a). Use a graph to guess the value of the limit (b). Use a graph of the sequence in part (a) to find the smallest values of N that correspond to ∈ = 0.1 and ∈ = 0.001 in Definition 3. lim - n!
> Sketch the region in the xy-plane. {(x, y) |x < 0}
> Find the slope and -intercept of the line and draw its graph. 4x + 5y = 10
> Find the slope and -intercept of the line and draw its graph. 3x – 4y = 12
> Find the slope and -intercept of the line and draw its graph. 2x – 3y + 6 = 0
> Find the slope and -intercept of the line and draw its graph. x + 3y = 0
> Find an equation of the line that satisfies the given conditions. Through (1/2, -2/3), perpendicular to the line 4x – 8y = 1
> Find an equation of the line that satisfies the given conditions. Through (-1, -2), perpendicular to the line 2x + 5y + 8 = 0
> Find an equation of the line that satisfies the given conditions. y-intercept 6, parallel to the line 2x + 3y + 4 = 0
> Find an equation of the line that satisfies the given conditions. Through (1, -6), parallel to the line x + 2y = 6
> Find an equation of the line that satisfies the given conditions. Through (4, 5), parallel to the -axis
> (a). For what values of x is it true that 1/x2 > 1,000,000 (b). The precise definition of limx→a f (x) = ∞ states that for every positive number M (no matter how large) there is a corresponding positive number δ such that if 0 < |x – a| < δ, then f (x) >
> Find the distance between the points. (1, -3) (5, 7)
> Find an equation of the line that satisfies the given conditions. Through (4, 5), parallel to the -axis
> Find an equation of the line that satisfies the given conditions. x-intercept -8, y-intercept 6
> Find an equation of the line that satisfies the given conditions. x-intercept, y-intercept -3
> Find an equation of the line that satisfies the given conditions. slope 3, y-intercept -2
> Find an equation of the line that satisfies the given conditions. Through (-1, -2), and (4, 4)
> Find an equation of the line that satisfies the given conditions. Through (2, 1), and (1, 6)
> Find an equation of the line that satisfies the given conditions. Through (-3, -5), slope -7/2
> Find an equation of the line that satisfies the given conditions. Through (2, -3), slope 6
> For the limit illustrate Definition 2 by finding values of N that correspond to ∈ = 0.5 and ∈ = 0.1. 4x2 + 1 lim = 2 x + 1
> Sketch the graph of the equation. |y| = 1
> Find the distance between the points. (1, 1), (4, 5)
> Rewrite the expression without using the absolute value symbol. |x2 + 1|
> Rewrite the expression without using the absolute value symbol. |2x – 1|
> Rewrite the expression without using the absolute value symbol. |x + 1|
> Rewrite the expression without using the absolute value symbol. |x – 2| if x > 2
> Rewrite the expression without using the absolute value symbol. |x – 2| if x < 2
> Show that if 0 < a < b, then a2 < b2.
> Prove that |ab| = |a||b|. [Hint: Use Equation 3.]
> Solve the inequality ax + b + c for x, assuming that a, b, and are negative constants.
> Use a graph to find a number N such that if x > N then |6х? + 5х — 3 3 < 0.2 2x2 – 1
> Solve the inequality a (bx – c) > bc for x, assuming that a, b, and are positive constants.
> Solve the inequality. |5x – 2| < 6
> Rewrite the expression without using the absolute value symbol. ||-2| - |-3||
> Solve the inequality. |2x – 3| < 0.4
> Solve the inequality. |x + 1|> 3
> Solve the inequality. |x + 5| > 2
> Solve the inequality. |x – 6| < 0.1
> Solve the inequality. |x – 4| < 1
> Solve the inequality. |x| > 3
> Solve the inequality. |x| < 3
> (a). How would you formulate an ∈1δ definition of the one-sided limit limx→a+ f (x) = L? (b). Use your definition in part (a) to prove that limx→a+ √x = 0.
> Solve the equation for x. |3x + 5| = 1
> Solve the equation for x. |x + 3| = |2x + 1|
> Rewrite the expression without using the absolute value symbol. |√5 – 5|
> Use the relationship between C and F given in Exercise 27 to find the interval on the Fahrenheit scale corresponding to the temperature range 20 < C < 30. Exercise 27: The relationship between the Celsius and Fahrenheit temperature scales is given by C
> The relationship between the Celsius and Fahrenheit temperature scales is given by C = 5/9 (F – 32), where C is the temperature in degrees Celsius and is the temperature in degrees Fahrenheit. What interval on the Celsius scale corresponds to the tempera
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 1/x < 4
> (a). Use Exercise 69 to show that the angle between the tangent line and the radial line is ψ = Ï€/4 at every point on the curve r = eθ. Exercise 69: Let P be any point (except the origin) on the curve r = f (Î&c
> Let P be any point (except the origin) on the curve r = f (θ). If ψ is the angle between the tangent line at P and the radial line OP, show that [Hint: Observe that ψ = φ - θ in the figure.]
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. (x + 1)(x – 2)(x + 3) > 0
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. x³ – x? < 0
> Identify the curve by finding a Cartesian equation for the curve. r= 3 sin e
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. x2 > 5
> Rewrite the expression without using the absolute value symbol. |π - 2|
> Use a graph to estimate the -coordinate of the highest points on the curve r = sin 2θ. Then use calculus to find the exact value.
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. x? < 2x + 8 2.1
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. (x – 1)(x – 2) > 0
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 1< 3x + 4 < 16
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 0 <1- x<1
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 1+ 5x > 5 – 3x
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 1 — х<2
> Show that the polar equation r = a sin θ + b cos θ, where ab ≠ 0, represents a circle, and find its center and radius.
> A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power.
> In this project we give a unified treatment of all three types of conic sections in terms of a focus and directrix. We will see that if we place the focus at the origin, then a conic section has a simple polar equation. In Chapter 10 we will use the pola
