For each function, find the amplitude, range, period, phase shift, vertical shift, and a sketch including a full period. a. f(t) = −sin(t − π2) + 3 b. f(x) = 4cos(2x + π2) − 1
The graph below models the monthly average precipitation in inches at a town on a particular mountain over a 3-year period, where x is the number of months after April 2020. a. Find the minimum average monthly precipitation. b. Find the maximum average monthly precipitation. c. Find the amplitude of the graph. d. Estimate the period and interpret its meaning. e. Use the function p(t) = acos(b(t − c)) + d to model the monthly average precipitation, p, as a function of time, t, measured in months.
A Ferris wheel is built such that the height h (in feet) above ground of a seat on the wheel at time t (in seconds) can be modeled by h(t) = 53 + 50sin(π10t − π2). a. Find the period of the model. What does the period tell you about the ride? b. Find the amplitude and the range of the model. What can we say about the ride? c. Find and interpret the average rate of change of h(t) on the time interval [2, 2.5].
Let tan(α) = 3/4 and cos(β) = −2/3, where α is in the first Quadrant and β is in the third Quadrant. a. Sketch and complete a right angle triangle for the angle α. b. Sketch and complete a right angle triangle for the angle β. c. Find sin(α + β). d. Find cos(α + β). e. Find tan(α + β). f. Find the quadrant containing the angle α+β.
If a person with weight W bends at the waist with a straight back, then the force F (in pounds) exerted on the lowe back muscles may be approximated by the function F(β) = 2.89Wsin(β + π/2). The angle β is the angle between a person's torso and the horizontal. a. Use the sum and difference identities to rewrite the function F(β) in terms of cosine. b. Suppose the person is standing straight, measure the force exerted on their back. c. Suppose the person weighs 200 -pounds i. Graph the function F(β) and describe the behavior of the function as the angle varies. ii. find the force exerted by the person's back muscles if β = 45∘, and iii. approximate the value of the angle β that results in the back muscles exerting a force of 350 pounds. Give an exact answer, then approximate it to two decimal places.
How many terms should be used to estimate the sum of the entire series with an error of less than 1/1000 ∑k = 1∞(−1)k+1 kk2+1
The improper integral ∫a∞f(x)dx will converge if A. the improper integral of g(x) where g(x) < f(x) also converges B. the improper integral of g(x) where g(x) > f(x) diverges C. the improper integral of g(x) where g(x) > f(x) also converges D. the improper integral of g(x) where g(x) < f(x) diverges E. None of the above
The infinite series ∑k = 1∞bk will converge if A. bk→L where 0 ≤ L < ∞ as k→∞ B. bk→0 as k→∞ C. bk→∞ as k→∞ D. bk+1 < bk for all k E. None of the above
What is a minimum and maximum value for the definite integral from x = 0 to x = 8 of the function shown in the graph? Do not solve the integral.
f(x) is integrable over the interval I = [a, b] only if A. df/dx is continuous on the interval I B. f(x) has only a finite number of jump discontinuities on the interval I C. df/dx has only a finite number of jump discontinuities on the interval I D. f(x) is continuous on the interval I E. None of the above