Math Solutions

Roll a die many times and take samples of size n. Show that the distribution of sample means approaches a normal distribution as n increases, regardless of the original distribution shape.

Find P(Z < 1.5) by shading the area under the standard normal curve to the left of z = 1.5. Show how the z-score converts any normal distribution to the standard normal.

Show the standard normal distribution N(μ, σ) with the 68-95-99.7 rule: 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. Adjust μ and σ to see how the curve shifts and spreads.

Identify and graph x² + 4xy + y² = 1 by rotating axes by angle θ = ½ arctan(B/(A−C)) to eliminate the xy term. Show the original and rotated coordinate systems.

Find and draw the tangent line to the ellipse x²/25 + y²/9 = 1 at the point (4, 9/5). Use implicit differentiation to find the slope, then write the tangent line equation.

Slice a double cone at different angles to produce a circle (horizontal cut), ellipse (tilted cut), parabola (cut parallel to edge), and hyperbola (steep cut through both cones). Show why they are called "conic sections."

Graph the plane 2x + 3y + z = 6 in 3D. Show the normal vector (2, 3, 1), the intercepts, and how changing the coefficients tilts the plane.

Find the distance between points (1, 2, 3) and (4, 6, 3) in 3D space using d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. Show the 3D coordinate system with the distance as the space diagonal of a rectangular box.

Plot x = A sin(at + δ), y = B sin(bt) for various frequency ratios a:b and phase shifts δ. Show the intricate patterns that emerge.

A point on the rim of a wheel of radius r rolls along a flat surface. Show that it traces a cycloid: x(t) = r(t − sin t), y(t) = r(1 − cos t). Animate the rolling wheel and the tracing point.

Plot x(t) = v₀ cos(θ) · t and y(t) = v₀ sin(θ) · t − ½gt² as a parametric curve. Show how the parameter t maps to position along the trajectory, with x(t) and y(t) displayed as separate time-domain graphs alongside the spatial path.

Graph r = cos(3θ) (three-petal rose), r = 1 + cos(θ) (cardioid), and r = 1 + 2cos(θ) (limaçon with inner loop). Show how the parameter n in r = cos(nθ) determines the number of petals.

Plot the points (3, π/4), (5, 2π/3), and (−2, π/6) in polar coordinates. Show the polar grid with concentric circles and radial lines. Convert between polar (r, θ) and Cartesian (x, y) coordinates.

Show how changing eccentricity e transforms a conic section: circle (e = 0) → ellipse (0 < e < 1) → parabola (e = 1) → hyperbola (e > 1). Use the focus-directrix definition: the ratio of distance-to-focus over distance-to-directrix equals e.

Show that every point on the parabola y = x² is equidistant from the focus (0, 1/4) and the directrix y = −1/4. Explore how changing the parameter p shifts the focus and widens or narrows the parabola.

Graph the hyperbola x²/16 − y²/9 = 1. Identify the vertices (±4, 0), asymptotes y = ±(3/4)x, and foci at (±5, 0). Show that for any point P on the hyperbola, |PF₁ − PF₂| = 2a = 8.

Draw the ellipse x²/25 + y²/9 = 1. Identify the semi-major axis a = 5, semi-minor axis b = 3, and foci at (±4, 0). Show that for any point P on the ellipse, the sum of distances PF₁ + PF₂ = 2a = 10 (the constant sum property).

Graph the circle (x − 3)² + (y − 2)² = 25. Identify the center (3, 2) and radius 5. Show how changing h, k, and r in (x − h)² + (y − k)² = r² moves and resizes the circle. Convert between standard form and general form x² + y² + Dx + Ey + F = 0.

Show the evolution of atomic models: Thomson's plum pudding (1904), Rutherford's nuclear model (1911), Bohr's orbits (1913), and the quantum mechanical electron cloud (1926). Explain what each model got right, what it got wrong, and what experiment forced the next model.

If 1 gram of matter is completely converted to energy, how much energy is released? Compare this to chemical explosions (TNT) and show the staggering scale of nuclear energy.