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Use Green’s Theorem to evaluate
∮C(x − 2y2) dx + (y4 + 2xy) dy where C consists of the line segment
from (0, 2) to (0, 4), followed by the curve with parametric equations x = 4 cos t, y = 4 sin t from (0, 4) to (−2, 2√3), then the line segment from (−2, 2√3) to (−1, √3), and finally the curve with parametric equations x = 2 sin t, y = 2 cos t from (−1, √3) to (0, 2).
Consider the equation xe^x = cos x
(a) Apply the intermediate value theorem to show that the function has a root in the interval
[0, 1].
Find the lengths of the sides of an isosceles triangle with a given perimeter if its area is to be as great as possible.
Use logarithmic differentiation to prove D9: d/dx (1/u^n) =( -n/u^n+1)(du/dx)
Equation of a Tangent Line: Find the standard (slope-intercept form) equation of the tangent line to the following functions at the specified points. (Show your complete solution, use Method 1)
- 𝑓(𝑥) = 𝑥2 − 4𝑥+1 at point (0,1)
- 𝑓(𝑥) = 𝑥3+ 2 at point (1,3)
- 𝑓(𝑥) = 2𝑥4+ 3𝑥3 − 2𝑥 + 7 at point (-1,8)
- 𝑓(𝑥) = √2𝑥 + 7 𝑎𝑡 𝑥 =1
MAXIMA, MINIMA, AND TIME RATES
Solve the following problems completely.
1. Find the positive number such that this number plus 36 times its reciprocal is a minimum.
2. Find two numbers whose sum is 24 such that the sum of the square of one plus six times the other is a minimum.
3. Find the dimensions of the largest rectangle that can be enclosed with 240m of fence.
4. A small jewelry box with square of base is to have a volume of 125 cu.cm. Find its dimensions to require the least amount of material.
5. A lot has the form of a right triangle, with perpendicular sides 60 and 80 feet long. Find the length and width of the largest rectangular building that can be erected, facing the hypotenuse of the triangle.
Find the volume generated by revolving the region y=x, y=x+2, y-axis about the line x = 4.
1) Find the area bounded by the curve y=9-x and the x-axis.
a) Horizontal Strip
b) Vertical Strip
2) Find the area bounded by the curve x = y² + 2y and the line x = 3.
3) Solve the area bounded by the curve y = 4x-x and the lines x = -2 and y = 4.
4) Find each of the two areas bounded by the curves y = x³ - 4x and y=x+ 2x.
Determine whether the given improper integrals are convergent or divergent. Evaluate those that are convergent.
Air is pumped into a spherical balloon at the rate of 100 cm3/s. How fast is the diameter increasing when the diameter is 10 cm?
The slope of f(x)=x3−12x2+36x−48 is zero at which value/s of x?
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