Question:
(a). Use Exercise 69 to show that the angle between the tangent line and the radial line is ψ = π/4 at every point on the curve r = eθ.
Exercise 69:
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
(b). Illustrate part (a) by graphing the curve and the tangent lines at the points where θ = 0 and π/2.
(c). Prove that any polar curve r = f (θ) with the property that the angle ψ between the radial line and the tangent line is a constant must be of the form r = Cekθ, where C and k are constants.
Transcribed Image Text:
tan dr/de
> 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 2/5, y-intercept 4
> 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
> 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
> An anticodon has the sequence GCG. What amino acid does this tRNA carry? What would be the effect of a mutation that changed the C of the anticodon to a G?
> Most of the carbon dioxide used in photosynthesis comes from _______. a. glucose b. the atmosphere c. rainwater d. photolysis
> A C3 plant absorbs a carbon radioisotope (as part of 14CO2). In which compound does the labeled carbon appear first? Which compound forms first if a C4 plant absorbs the same radioisotope?
> 1. Photosynthesis runs on the energy of ______. a. light b. hydrogen ions c. O2 d. CO2 2. In cyanobacteria and photosynthetic eukaryotes, the light-dependent reactions proceed in/at the ______. a. thylakoid membrane b. plasma membrane c. stroma d. cytop
> While gazing into an aquarium, you see bubbles coming from an aquatic plant (right). What are the bubbles?
> A cat eats a bird, which ate a caterpillar that chewed on a weed. Which organisms are autotrophs? Which ones are heterotrophs?
> _______ cannot easily diffuse across a lipid bilayer. a. Water b. Gases c. Ions d. all of the above
> All antioxidants ______. a. prevent other molecules from being oxidized b. are coenzymes c. balance charge d. deoxidize free radicals
> A molecule that donates electrons becomes ______, and the one that accepts electrons becomes ______. a. reduced; oxidized b. ionic; electrified c. oxidized; reduced d. electrified; ionic
> Immerse a human red blood cell in a hypotonic solution, and water _______. a. diffuses into the cell b. diffuses out of the cell c. shows no net movement d. moves in by endocytosis
> What accumulates inside the thylakoid compartment of chloroplasts during the light-dependent reactions? a. glucose b. hydrogen ions c. O2 d. CO2
> On the geologic time scale, life originated in the _______. a. Archaean b. Proterozoic c. Phanerozoic d. Cambrian
> A chromosome contains many different gene regions that are transcribed into different __________. a. proteins b. polypeptides c. RNAs d. a and b
> In the late 1970s, geologist Walter Alvarez was investigating the composition of the K–Pg boundary layer in different parts of the world. He asked his father, Nobel Prize–winning physicist Luis Alvarez, to help him ana
> The number of species on an island depends on the size of the island and its distance from a mainland. This statement would most likely be made by __________. a. an explorer b. a biogeographer c. a geologist d. a philosopher
> In the late 1970s, geologist Walter Alvarez was investigating the composition of the K–Pg boundary layer in different parts of the world. He asked his father, Nobel Prize–winning physicist Luis Alvarez, to help him ana
> Name one environmental factor that typically influences enzyme function.
> Dogs have a diploid chromosome number of 78. How many chromosomes do their gametes have? a. 39 b. 78 c. 156 d. 234
> Meiosis ______ the parental chromosome number. a. doubles b. halves c. maintains d. mixes up
> Crossing over happens during which phase of meiosis? a. prophase I b. prophase II c. anaphase I d. anaphase II
> Crossing over mixes up _______. a. chromosomes b. alleles c. zygotes d. gametes
> When DNA replication begins, ______. a. the two DNA strands unwind from each other b. the two DNA strands condense for base transfers c. old strands move to find new strands
> 1. Meiosis is a necessary part of sexual reproduction because it ______. a. divides two nuclei into four new nuclei b. reduces the chromosome number for gametes c. gives rise to new alleles 2. Meiosis ______. a. occurs only in animals b. supports growth
> One evolutionary advantage of sexual over asexual reproduction is that it produces ______. a. more offspring per individual b. more variation among offspring c. healthier offspring
> Researchers are designing and testing antisense drugs as therapies for a variety of diseases, including cancer, AIDS, diabetes, and muscular dystrophy. The drugs are also being tested to fight infection by deadly viruses such as Ebola. Antisense drugs co
> The diploid chromosome number for the body cells of a frog is 26. What would that number be after three generations if meiosis did not occur before gamete formation?
> ______ are always changed by participating in a reaction. (Choose all that are correct.) a. Enzymes c. Reactants b. Cofactors d. Coenzymes
> Catalase combines two hydrogen peroxide molecules (H2O2 + H2O2) to make two molecules of water. A gas also forms. What is the gas?
> ___ is life’s primary source of energy. a. Food b. Water c. Sunlight d. ATP
> The probability of a crossover occurring between two genes on the same chromosome _______. a. is unrelated to the distance between them b. decreases with the distance between them c. increases with the distance between them
> Refer to question 4. Assuming complete dominance, the F2 generation will show a phenotypic ratio of _______. a. 3:1 b. 9:1 c. 1:2:1 d. 9:3:3:1
> F1 offspring of the cross AA × aa are ________. a. all AA b. all aa c. all Aa d. 1/2 AA and 1/2 aa
>  The graph shown in FIGURE 8.5 is reproduced from an original 1952 publication by Hershey and Chase. Bacteriophage were labeled with radioactive tracers and allowed to infect bacteria. The virus–bacte
> An organism’s observable traits constitute its _______. a. phenotype b. variation c. genotype d. pedigree
> The enzyme trypsin is sold as a dietary enzyme supplement. Explain what happens to trypsin taken with food.
