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mathematics
calculus

Questions and Answers of Calculus

One player makes 5 out of 10 shots, another makes 16 out of 20. Use the method of support to test whether the above samples differ.
A 1 m2 region in Utah is hit by 4 cosmic rays in 1 yr, and a 1 m2 region at the North Pole is hit by 10 cosmic rays in 1 yr. Use the method of support to test whether the above samples differ.
A coin is flipped 10 times and comes up heads 9 times. Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5.
Two 1 m2 regions in Utah are hit by 3 and 5 cosmic rays in 1 yr, and a 1 m2 region at the North Pole is hit by 10 cosmic rays in 1 yr. Use the method of support to test whether the above samples
One player makes 5 out of 10 shots, another makes 9 out of 10. Use the G test to test the above by building a table complete with observed and expected values. Compare the G statistic with the
As in Exercise 28, one player makes 5 out of 10 shots, another makes 16 out of 20. Use the G test to test the above by building a table complete with observed and expected values. Compare the G
Check whether organisms 1 and 2 differ and compare with Section 8.6, Exercise 39.One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28 in 1 million.
Check whether organisms 2 and 3 differ and compare with Section 8.6, Exercise 40.One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28 in 1 million.
A cell in the salubrious bath survives 30 min, and a cell in standard culture survives only 5 min. Is there reason to think that the salubrious bath lengthens cell life? It has been proposed that a
In a repeated experiment, the cell in the salubrious bath survives 60 min, and a cell in standard culture survives only 3 min. Is there reason to think that the salubrious bath lengthens cell
Combine the data from Exercises 35 and 36, and evaluate the difference in support between the null and alternative hypotheses. It has been proposed that a particular salubrious bath extends cell
What would happen to the result in Exercise 37 if a third experiment were done and both cells survived 10 min? Why the change? It has been proposed that a particular salubrious bath extends cell
A coin is flipped 20 times and comes up heads 3 times. Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5.
One cosmic ray hits a detector in 1 yr. The null hypothesis is that the rate at which rays hit is λ = 5/yr. Use the method of support to evaluate the above null hypotheses.
Three cosmic rays hit a larger detector in 1 yr. The null hypothesis is that the rate at which rays hit is λ = 10/yr. Use the method of support to evaluate the above null hypotheses.
You wait 4000 h for an exponentially distributed event to occur. The null hypothesis is that the mean wait is 1000 h with alternative that the mean wait is greater than 1000 h. Find the difference in
You wait 40 h for an exponentially distributed event to occur. The null hypothesis is that the mean wait is 1000 h with alternative that the mean wait is less than 1000 h. Find the difference in
The first defective gasket is the 25th. The null hypothesis follows a geometric distribution with mean wait 10, and the alternative is that the mean wait is greater than 10. Find the difference in
Plot yield (Y) against weight (W). Suppose we think that the line Y = W + 0.6 describes these data. Plot the line on your graph of yield against weight. Find and plot the residuals.Consider the data
Compute the best fitting line for height as a function of weight. Graph the line.Consider the data in the following table.
Find the correlation between weight and yield. Check that its square is equal to the value of r2 found in Exercise 11.Consider the data in the following table.
Find the correlation between weight and height. Check that its square is equal to the value of r2 found in Exercise 12.Consider the data in the following table.
Find the best fitting line and r2 for replicate 1 and then test whether the diet has a significant effect.Linear regression has important connections with other techniques in statistics, such as
Find the best fitting line and r2 for replicate 2 and then test whether the diet has a significant effect. Compare with the previous problem.Linear regression has important connections with other
Show that the best linear fit from Theorem 8.6 passes through the center of the data in the sense that the sum of the residuals is 0. Best fit regression lines have many nice properties.
Consider models of the form Y = b. Show that the sum of the squares of the residuals is minimized when b = , the sample mean of the yi. Best fit regression lines have many nice properties.
Plot height (H) against weight (W). Suppose we think that the line H = 8W + 5 describes the data. Plot the line on your graph of height against weight. Find and plot the residuals.Consider the data
Consider models of the form Y = aX. Find the slope that minimizes the sum of the squares of the residuals. Best fit regression lines have many nice properties.
Find the best linear fit. Plot the line and find r2. How good is the model? Consider the following measurements. x y 1.0 ........... 1.1 2.0 ........... 3.9 3.0 ........... 8.8 4.0
Use the principle of least squares to write the expression you would use to fit a curve of the form Y = aX2 + b. One easy way to solve this is to think of a new measurement Z = X2 and find the linear
Check that the dimensions for each term of the regression equation for toxin tolerance as a function of mass (Example 8.9.2) are consistent.1. Find the dimensions of the slope â.2. Find the
Find r2 for the line bt+1 = 2bt. Graph the data and the line Consider the following data describing change in a bacterial population.
Find the best fitting line, and compare with a mathematically idealized model. Which makes more sense?Consider the following data describing change in a bacterial population.
Find the best fitting line for population 1 as a function of time and compute r2.Consider the following data on the growth of two bacterial populations.
Find the best fitting line for population 2 as a function of time and compute r2.Consider the following data on the growth of two bacterial populations.
Find the best fitting line for the logarithm of population 1 as a function of time and compute r2. Is this a better fit?Consider the following data on the growth of two bacterial populations.
Find the best fitting line for the logarithm of population 2 as a function of time and compute r2. Is this a better fit?Consider the following data on the growth of two bacterial populations.
Find the best fitting line and r2 for replicate 1 with and without the fourth point. Graph the two regression lines and the data.Consider the following data which include one outlying point. Find the
Find the best fitting line and r2 for replicate 2 with and without the last point. Graph the two regression lines and the data. Why do you think the outlier affects this regression line more?Consider
Compute the best fitting line for yield as a function of weight. Graph the line.Consider the data in the following table.
Use a calculator to find a rational number r such that |r − π2| < 10−4.
Express the set of numbers x satisfying the given condition as an interval. |x − 4| < 2
Express the set of numbers x satisfying the given condition as an interval. |4x − 1| ≤ 8
Describe the set as a union of finite or infinite intervals. {x : |x − 4| > 2}
Describe the set as a union of finite or infinite intervals. {x : |x - 4| > 2}
Match (a)-(f) with (i)-(vi).(a) a > 3(b) |a ˆ’ 5| (c)(d) |a| > 5(e) |a -4| (f) 1 ‰¤ a ‰¤ 5(i) a lies to the right of 3.(ii) a lies between 1 and 7.(iii) The distance from a to 5 is
Describe {x : x2 + 2x < 3} as an interval
Show that if a > b, then b-1 > a -1, provided that a and b have the same sign. What happens if a > 0 and b < 0?
Show that if | a - 5| < 1/2 and |b -8|
Give an example of numbers a and b such that |a + b| < |a| + |b|.
Suppose that |a − 6| ≤ 2 and |b| ≤ 3. (a) What is the largest possible value of |a + b|? (b) What is the smallest possible value of |a + b|?
Express r1 = 0.27 as fraction. 100r1 - r1 is an integer. Then express r2 = 0.2666 . . . as a fraction.
The text states: if the decimal expansions of numbers a and b agree to k places, then | a - b | ≤ 10−k. Show that the converse is false: For all k there are numbers a and b whose decimal
Find the equation of the circle with center (2, 4): (a) With radius r = 3. (b) That passes through (1,−1).
Determine the domain and range of the function f : {r, s, t , u} → {A,B,C,D,E} defined by f (r) = A, f (s) = B, f (t) = B, f (u) = E.
Find the domain and range of the function. 1. f (x) = −x 2. f (x) 4 = x3 3. f (x) = |x| 4. f (x) = 1/x2
Determine where f (x) is increasing 1. f(x) = |x + 1| 2. f(x) = x4
In which quadrant do the following points lie? (a) (1, 4) (b) (−3, 2) (c) (4,−3) (d) (−4,−1)
Find the zeros of f (x) and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.1. (x) = x2 - 42. f(x) = x3 − 4x3. f(x) = 2 − x3
Which of the curves in Figure 26 is the graph of a function?
Express the interval in terms of an inequality involving absolute value. (0, 4)
Determine whether the function is even, odd, or neither.(a)(b) g(t) = 2t - 2-t (c) G (Î¸) = sin Î¸ + cos Î¸ (d) H (Î¸ = sin(Î¸2)
Determine the interval on whichIs increasing or decreasing.
Let f (x) be the function shown in Figure 27.1. Find the domain and range of f (x)? 2. Sketch the graphs of f (2x), f(1/2x), and 2f (x). 3. Extend the graph of f (x) to [ˆ’4, 4] so that it
Express the interval in terms of an inequality involving absolute value. [1,5]
Suppose that f (x) has domain [4, 8] and range [2, 6]. Find the domain and range of: (a) f (x) + 3 (b) f (x + 3) (c) f (3x) (d) 3f (x)
Suppose that the graph of f (x) = sin x is compressed horizontally by a factor of 2 and then shifted 5 units to the right. (a) What is the equation for the new graph? (b) What is the equation if you
Sketch the graph of f (2x) and f(1/2x), where f (x) = |x| + 1.
Define f (x) to be the larger of x and 2 − x. Sketch the graph of f (x). What are its domain and range? Express f (x) in terms of the absolute value function.
Show that the sum of two even functions is even and the sum of two odd functions is odd.
Prove that the only function whose graph is symmetric with respect to both the y-axis and the origin is the function f (x) = 0.
Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if b = 2n5m for some n,m ≥ 0.
A function f (x) is symmetric with respect to the vertical line x = a if f (a − x) = f (a + x). (a) Draw the graph of a function that is symmetric with respect to x = 2. (b) Show that if f (x) is
Find the slope, the y-intercept, and the x-intercept of the line with the given equation. 1. y = 3x + 12 3. 4x + 9y = 3
Find the equation of the line with the given description 15. Parallel to y = 3x − 4, passes through (1, 1) 17. Perpendicular to 3x + 5y = 9, passes through (2, 3) 19. Horizontal, passes through (8,
Are the lines y = 2x + 1 and y = −2x − 4 perpendicular?
Find the equation of the perpendicular bisector of the segment joining (1, 2) and (5, 4) (Figure 11). Hint: The midpoint Q of the segment joining (a, b) and (c, d) is
Determine whether there exists a constant c such that the line x + cy = 1: (a) Has slope 4 (b) Passes through (3, 1) (c) Is horizontal (d) Is vertical
Materials expand when heated. Consider a metal rod of length L0 at temperature T0. If the temperature is changed by an amount Δ T , then the rod's length changes by ΔL = αL0ΔT , where α is the
Find b such that (2,−1), (3, 2), and (b, 5) lie on a line
The period T of a pendulum is measured for pendulums of several different lengths L. Based on the following data, does T appear to be a linear function of L?
Find the roots of the quadratic polynomials: (a) 4x2 − 3x - 1 (b) x2 − 2x - 1
Complete the square and find the minimum or maximum value of the quadratic function. 35. y = x2 − 6x + 9 37. y = x2 + 6x + 2
Complete the square and find the minimum or maximum value of the quadratic function. 39. y = −4x2 + 3x + 8 41. y = 4x − 12x2
Sketch the graph of y = x2 + 4x + 6 by plotting the minimum point, the y-intercept, and one other point.
For which values of c does f (x) = x2 + cx + 1 have a double root? No real roots?
Prove that x + 1/x ≥ 2 for all x > 0. Consider (x1/2 − x−1/2)2.
If objects of weights x and w1 are suspended from the balance in Figure 13(A), the cross-beam is horizontal if bx = aw1. If the lengths a and b are known, we may use this equation to determine an
Find the slope of the line. 5. y = 3x + 2 7. 3x + 4y = 12
Find a pair of numbers whose sum and product are both equal to 8.
Show that if f (x) and g(x) are linear, then so is f (x) + g(x). Is the same true of f (x)g(x)?
Show that Δy/Δx for the function f (x) . = x2 over the interval [x1, x2] is not a constant, but depends on the interval. Determine the exact dependence of Δy/Δx on x1 and x2.
Let a, c ≠ 0. Show that the roots of ax2 + bx + c = 0 and cx2 + bx + a = 0 are reciprocals of each other.
Prove Viète's Formulas: The quadratic polynomial with α and β as roots is x2 + bx + c, where b = −α − β and c = αβ
Find the equation of the line with the given description. 9. Slope 3, y-intercept 8 11. Slope 3, passes through (7, 9) 13. Horizontal, passes through (0,−2)
Give an example of a rational function.
Determine the domain of the function. 1. f (x) = x1/4 3. f (x) = x3 + 3x - 4 5. g(t) = 1/t + 2
Identify each of the following functions as polynomial, rational, algebraic, or transcendental. 13. f(x) = 4x3 + 9x2 - 8 15. f (x) =√x 17. f (x) = x2 /x + sin x
Identify each of the following functions as polynomial, rational, algebraic, or transcendental. 19. f(x) = 2x3 + 3x/9 − 7x2 21. f (x) = sin(x2) 23. f(x) = x2 + 3x−1
Calculate the composite functions f ◦ g and g ◦ f, and determine their domains. 27. f (x) =√x, g(x) = x + 1 29. f(x) = 2x , g(x) = x2

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