Theorem Explanation
Justify whether the Extreme Value Theorem can be applied or not in the given interval.
f(x) = (x2+3x+6)1/3 + sin2(x2), [2,0]
EVT Condition: Continuous on [a,b]
(x2+3x+6)1/3 & sin2(x2) are both continuous on all x, so yes, EVT can be applied to f(x).
Solve:
lim (3x4 + 5x3 + 3x2 / x3 + 10x2)x->0
3/20
Given the position function, s(t) = 4cos(0.5t) + 0.25t2, find out the position of the object when its acceleration is zero in the interval of 0 ≤ t ≤ pi. Round to the thousandth place
3.097
A rectangular fence has to be built adjacent to the house to make a garden for the dog to run around. The area must cover 500,000m2. What dimension of fencing would require the least amount of fencing if one side is not fenced?
500m x 1000m
Angelina is trying out for the Bobsleigh Olympics. She can be qualified if she is able to reach a speed of 85 km/hr at least once during her race. On the track, camera A on the track detects that his speed is 78km/hr. Three minutes later, camera B, which is 4.2km away, detects that his speed is 82km/hr. Can Angelina use the Mean Value Theorem to prove that her speed went over the required 85km/hr? Show your work.
MVT proves that Angelina's speed reached 84km, so it can not be used to guarantee that Angelina qualified for the olympics.
Using linear approximation with f(x) = (x+3)1/5 at x=-2, find the value of f(-2.5). Find the error.
L(x) = 1/5 (x+2) +1
L(-2.5) = 0.9
f(-2.5) = 0.8705...
Error = 0.029 ; 10-1
A calculator manufacturer produces graphing calculators for the calculus students suffering all around the world. Find the quantity x that maximizes profit if the total revenue, R(x) and the total cost C(x) are…
R(x) = 9x - 0.0025x2
C(x) = 1500 + 2.5x
1300 Calculators
What is the maximum area of the rectangle inscribed in a semicircle with a radius of 10 inches?
100in2
Does the EVT apply? If so, find the absolute maximum. If not, justify why.
f(x) = ex^3+6x^2+12x on the interval [-3,1]
EVT applies because f(x) is continuous on [-3,1]
Abs. max @ x=1, f(1) = 178,482,301
Using linear approximation, estimate y at x=0.95, given the implicit curve:
y5 + ln(x/y2) +2x2 = 3, and that point (1,1) exists on the curve.
13/12
A lamppost is 38ft above the ground. Able is a 6ft tall, handsome, popular college student, and is walking away from the lamp at a rate of 5ft per second. When he is 15ft away from the lamppost, how fast is the length of his shadow changing?
His shadow changes at a rate of 15/16 ft/sec
A zipline is being installed from an island 10 miles off the coast, to a hill 25 miles down the coast. The cost to lay a zipline is $53,000 per mile above the ocean and $35,000 per mile along the shoreline. How should the zipline be installed to minimize the cost if assumed that the zipline is installed perfectly parallel to the ground? Round to the thousandth place if necessary.
8.794mi