BiologyCell Biology
What Are the Monomers of Lipids?
Learn why glycerol and fatty acids are the building components of fats, while lipids have no universal repeating monomer.
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BiologyCell Biology
How Many ATP Are Produced in Glycolysis? Total vs. Net
Glycolysis produces 4 ATP per glucose in total and gains 2 ATP net after its investment. See the accounting and pathway boundary.
BiologyGenetics
Where Does Transcription Take Place in a Eukaryotic Cell?
The nucleus is the expected answer for nuclear genes, but mitochondria and chloroplasts also transcribe their own DNA.
MathCalculus
Integral Test for Convergence: Does ∑ n/(3n² + 1) Converge?
The positive decreasing extension has a divergent logarithmic improper integral, so ∑ n/(3n² + 1) diverges by the integral test.
MathCalculus
Shell Method Formula: Volume of y = 3x − x² About the y-Axis
Using shells of radius x and height 3x − x² gives V = 2π∫₀²x(3x − x²) dx = 8π cubic units.
MathCalculus
Derivative of Trigonometric Functions: Differentiate csc x + x tan x
The sum and product rules give f′(x) = −csc x cot x + tan x + x sec²x on the original function's domain.
MathCalculus
What Is the Mean Value Theorem? Apply It to a Falling Rock
The rock hits after 5/2 s, its average velocity is −40 ft/s, and the Mean Value Theorem identifies c = 5/4 s when instantaneous velocity matches it.
PhysicsClassical Physics
Elastic vs Inelastic Collision: Which Cases Conserve Kinetic Energy?
The first record is perfectly inelastic because the objects stick and kinetic energy falls from 27 J to 24 J. Records 2 and 3 conserve both momentum and kinetic energy, so they are perfectly elastic.
PhysicsClassical Physics
How to Find Tension in a Two-Mass Atwood Machine
For ideal masses of 2.00 kg and 4.00 kg, the heavier mass accelerates downward at 3.27 m/s², the lighter mass accelerates upward at the same rate, and the string tension is 26.1 N.