
Concept explainers
Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the
We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps:
1. Express the height h and the base b of the isosceles triangle in Figure 2.31 in terms of

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Chapter 2 Solutions
Calculus Volume 1
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University Calculus: Early Transcendentals (4th Edition)
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Elementary Statistics: Picturing the World (7th Edition)
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Calculus: Early Transcendentals (2nd Edition)
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- The U.S. Bureau of Labor Statistics reports that 11.3% of U.S. workers belong to unions (BLS website, January 2014). Suppose a sample of 400 U.S. workers is collected in 2014 to determine whether union efforts to organize have increased union membership. a. Formulate the hypotheses that can be used to determine whether union membership increased in 2014.H 0: p H a: p b. If the sample results show that 52 of the workers belonged to unions, what is the p-value for your hypothesis test (to 4 decimals)?arrow_forwardFind a unit normal vector to the surface f(x, y, z) = 0 at the point P(-3,4, -32) for the function f(x, y, z) = In -4x -5y- Please write your answer as a vector (a, b, c) with a negative z component, and show your answer accurate to 4 decimal placesarrow_forwardAnswer this pleasearrow_forward
- B1 The x distribution is a special case of Gamma distribution (not to be confused with gamma function; see below). The density function of the Gamma distribution with parameters and k is given by where -1 e -x/0 (x) = = г(k) Øk if x > 0, and otherwise, г(k) = √ ₁ k-1-x dx x' e is the gamma function. (a) For every k ≥ 1, 0 > 0, find the mode of the density. Hint: The algebra can be simplified by appropriate use of logarithms. ~ Now suppose that X1,..., Xn id Exp(\) and that we have a prior belief in A which is consistent with a prior distribution X. Gamma(a, b), for some a, ß, i.e. the prior density of is Baxa-1-BA T(a) e = so 01/ẞ and k = a. (b) Write down the likelihood, and show that the posterior distribution for \ is also a Gamma distribution, but with parameters a +n and B + Σ Xi. (c) Find the mode of the posterior distribution and examine the behaviour as n → ∞.arrow_forwardFind the differential of the function f(x, y) = 2x² - 2xy – 5y² at the point (-6, -5) using Ax = 0.3 and Ay = 0.05. dz = Now find Az and compare it to your answer above Ax= Hint: If entering a decimal, round to at least 5 placesarrow_forwardFind the differential of the function f(x, y) = −8x√y at the point (1,3) using Ax = 0.25 and Ay = -0.15. dz Now find Az and compare it to your answer above Az = Hint: If entering a decimal, round to at least 5 placesarrow_forward
- please dont use chat gpt i need to underarrow_forwardChris Lynch plans to invest $200 into a money market account. Find the interest rate that is needed for the money to grow to $1,800 in 12 years if the interest is compounded quarterly. The rate is %. (Round to the nearest percent.)arrow_forwardFind the interest earned on $10,000 invested for 6 years at 6% interest compounded as follows. a. Annually b. Semiannually (twice a year) c. Quarterly d. Monthly e. Continuouslyarrow_forward
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