Calculus Answers

Express the function 𝑓(𝑥) = 𝑥 /(𝑥−3)² as partial fractions and hence find ∫ 𝑓(𝑥)𝑑x

A function is defined by the polynomial 𝑓(𝑥) = 3𝑥⁴ − 4𝑥³ − 12𝑥² + 8. Find and classify all the stationary points of f(x).

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 ∫ 𝑥𝑐𝑜𝑠 𝑥𝑑x

The region above the x-axis and below the curve 𝑦 = 𝑒 −𝑥 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ ∞ is rotated around the x-axis. Find the volume of the solid generated.

Express the function 𝑓(𝑥) = 𝑥 (𝑥−3)2 as partial fractions and hence find ∫ 𝑓(𝑥)𝑑�

Show that 𝑑𝑦 𝑑𝑥 = 𝑠𝑒𝑐2𝑥 given that 𝑦 = tan x

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

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)