Calculus Answers

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? 

e^x

d/ dx[ ∫ (Int) dt] = x-In 2

2

True or false with full explanation

Differentiate 5x.Sin(4πx)

Given that z= x² + y² + e^t, y=sin(x), x= cos(t). Find dz/dt

Given that 𝑧1 = 3 + 𝑖 𝑎𝑛𝑑 𝑧2 = 2 − 𝑖: i. Find the modulus and argument of 𝑧1/𝑧2 (5 marks) ii. Express 𝑧1/𝑧2 in polar and exponential form (2 marks) iii. Use de Moivre’s theorem to find an expression for ( 𝑧1/𝑧2) ^4 

Evaluate lim𝑥→2 𝑥^3−2𝑥^2+4𝑥−8/x^4−2𝑥^3+𝑥−2

A function is defined by the polynomial 𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 − 12𝑥 2 + 8. Find and classify all the stationary points of f(x).

For what value of b is the function 𝑓(𝑥) = { 𝑥 2 − 1 𝑥 < 3 2𝑏𝑥 𝑥 ≥ 3 continuous everywhere

Show that 𝑑𝑦/𝑑𝑥 = 𝑠𝑒𝑐^2 x given that 𝑦 = tan x

Given that 𝑓(𝑥) = 3𝑥 2 − 4𝑥 + 7, use the definition of the derivative to find 𝑓 ′ (𝑥)