Calculus Answers

Find ∫ ln 3𝑥 − 1 𝑥 5 + cos 4𝑥 − 𝑒 −𝑥 2 ⁄ 𝑑x

Calculate the work done by a Force of 2𝑖 − 𝑗 − 3𝑘 in moving an object from (2, 1, 3) to (9, 4, 6) where displacement is in metres.

Given the system of equations 2 3 2 11 3 2 3 7 4 4 14 x y z x y z x y z + − = − − + = − + = , find the values of x, y, and z using matrix inversion. 

Consider the equation xyz = 4x² +y² −z². Use the Implicit Function Theorem to show that the given equation has a smooth unique local solution of the form z = g(x,y) about the point (2,0,4). Then find the local linearization of g about the point (2,0).

As a phase line, use a U-shaped curve. Let its downward-sloping segment intersect the 45 degree line at point L, and let its upward-sloping segment intersect the 45° line at point R. Answer the following five questions:

(a) Is this a case of multiple equilibria?

(b) If the initial value y₀ lies to the left of L, what kind of time path will be obtained?

(c) What if the initial value lies between Land R?

(d) What if the initial value lies to the right of R?

(e) What can you conclude about the dynamic stability of equilibrium at L and at R, respectively?                                               

Find the extreme values of f(x,y)=x+y^2 on the surface 2x^2+y^2=1.

Lim_x→0 (sin x -x) / tan x

Lim_x→∞ -(x+1)(e^1/(x+1) -1)

welch construction has found that the value of the company tractor is decreasing at a rate give by v(t)= -390te^{-0.012t^2}, where t represents the number of years after purchase. if the company purchased the tractor for $17650, what is the value of the tractor 3 years after purchase

Derive the reduction formula

integeration(x²+a²)^n/2dx = x(x²+a²)^n/2÷n+1+ na²/n+1×integeration (x² +a²)^n/2-1dx

Use the formula to integrate integration(x²+a²)^5/2