### Calculus III

Please show all the steps to solving this problem.

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Please show all the steps to solving this problem.

What is the equation of the plane determined by the three points A(1,0,2), B(3,-1,6), C(5,2,4)?

Coordinate geometry and calculus. (See the attached file for the questions).

Find the stationary points, stating their position & type of the following equation of a curve. Y = X³ - 7.5X² + 18X + 29

An object moves along the x-axis with initial position x(0)=2. The velocity of the object at time t is greater than or equal to 0 is given by v(t)=sin((pi/3)t). a.) What is the acceleration of the object at time t=4? b.) Consider the following two statements. Statement I: For 3<t<4.5, the velocity of the object is increa

The function f(x)=x^4 - 4x^3 + 8x has a critical point at x=1. Use the second derivative test to identify it as a local maximum, a local minimum, or neither.

The following statements are either true or false. a). If x=p is not a critical point of f, then x=p is not a local maximum of f. b). If x=p is not a local maximum of f, then x=p is not a critical point of f. c). If f'(x) is continuous and f(x) has no critical points, then f is everywhere increasing or everywhere decrea

1) Solve the following inequality 2) Graph it sqrt(x+6) / x^2 ≥ 1/x

Hi, I attached one problem. Please help me providing detailed solution with explanation. Thanks

Please follow all instructions and answer accordingly. See the attached file. Consider a projectile of mass m

Please give all possible explanations. See attached. Solve the following differential equation using the complex number approach.

Consider a projectile of mass m which is shot vertically upward from the surface of the earth with initial velocity V. Assume that the gravitational force acts downward at a constant acceleration g while the force of air resistance has a magnitude proportional to the square of the velocity with proportionality constant k>0 and a

Hi, I attached the problem. Please help me providing detailed solution with explanation.

Find the Interval of Convergence (include work for endpoints) for ∑∞k=0 [1/(sqrt(k^2 + 1))] * x^k This is the Riemann sum from 0 to infinity (see attached)

Let f(x) = (x-5)/x Use calculus theorems/techniques to show that the equation has exactly one solution in the interval [1, 8]. Explain carefully. Confirm that f satisfies the hypotheses of the Mean Value Theorem on the interval [1, 8] and find all numbers c in this interval which satisfy the conclusion of the theorem. Show w

Find the Jacobian of the transformation x= u + v + w, y= u + v - w, z= u - v + w.

Please see the attached file.

Find the limit, if it exists. Limit (x, y -> 0) f(x) = 15 x^3 y /(2x^4 + y^4) Please see the attached file.

Please see the attached file. dy/dx = 3/y First find a general solution of the differential equation. Then find a particular solution that satisfies the initial condition y(0) = 5. ******************* A bacteria population is increasing according to the natural growth formula and numbers 100 at 12 noon and 156

Find the interval where the function is increasing and the interval where it is decreasing. (If you need to enter - or , type -INFINITY or INFINITY. If there is no interval where the function is increasing/decreasing, enter NONE in those blanks.) ( , ) (increasing) ( , ) (decreasing) 2. [TanApCalc7 4.1.0

Please see the attached file for the fully formatted problems. Let f(x) = 1 + (a) For what values of x is f defined? (b) Find a power series for xf'(x). Justify your answer. (c) Hence show that: f(x) = 1 -

1. Give the value of c in the interval [0,3] that satisfies the conclusion of the mean value theorem for -2*sqrt(x)-2 2. Use differentials to estimate the fourth root of 80. 3. Use differentials to estimate the sine of 44 degrees. 4. Use one iteration of Newton's method to approximate a solution to x^3+x-352=0 with initia

Find the exact length of the curve. x = 7cos(t) - cos(7t) y = 7sin(t) - sin(7t) 0<= t < = pi;

r = 10sin(theta) + 10cos(theta) which one is it ? circle ellipse parabola line

Please help me to solve attached problem. 1. Determine whether one of the solutions of the differential equation x^2y" + xy' + (x^2 -1/4)y =0 on 0<x<infinity is y(x) = x^-1/2sinx

Suppose that p and q are continuous on some open interval I, and suppose that y1 and y2 are solutions of the ODE y'' + p(t)y' + q(t)y = 0 on I. (a) Suppose that y1, y2 is a fundamental set of solutions. Prove that z1, z2, given by z1 = y1 + y2, z2 = y1 − y2, is also a fundamental set of solutions. (b) Prove that i

Solve Laplace of u = 0 subject to the conditions: u(x,0) = f1(x) u(0,y) = 0 u(x,b) = 0 u(a,y) = 0 0<x<a 0<y<b (The question attachment contains a slightly different question. The question is restated correctly in the solution attachment)

It is essential to show all steps by hand!! Also if a method is prescribed use only that method! 8)justifying its use, use L'Hospital's Rule to evaluate the following limit: lim ln(x^8) / 8x^8-8 x->1

Find the upper and lower control limits for /x- and R-charts for the width of a chair seat when the sample grand mean (based on 30 samples of 6 observations each) is 27.0 inches, and the average range is 0.35 inches.

Use one-sided limits to find the limit or determine that the limit does not exist: limx => 4 16-x^2 / 24-x The problem might be easier to read like this: limit 16 - x^2 x --> 4 24 - x