Math Solutions

Graph y = sin(x) and y = cos(x) on the same axes from 0 to 2π and study their relationship.

Visualize sin(θ) and cos(θ) as a point moves around the unit circle.

If F(x) = ∫₀ˣ t² dt, show that F'(x) = x².

A spring with k = 100 N/m is compressed by 0.5 m. Calculate the work done.

Find the average value of f(x) = sin(x) on the interval [0, π].

If water flows into a tank at rate r(t) = 5t - t² liters/min, how much water enters from t = 0 to t = 5?

Find the length of the curve y = x² from x = 0 to x = 2.

Rotate the region between y = x and y = x² around the x-axis from x = 0 to x = 1.

Rotate y = √x from x = 0 to x = 4 around the x-axis. Find the volume of the resulting solid.

Find the area between y = 2x and y = x² from x = 0 to x = 2.

Show how the area approximation under y = x² from 0 to 4 converges as n increases from 4 to 100.

Approximate ∫₀⁴ x² dx using 8 midpoint rectangles.

Approximate ∫₀⁴ x² dx using 8 right-endpoint rectangles.

Approximate the area under y = x² from 0 to 4 using 8 left-endpoint rectangles.

You have 100 m of fence. What rectangle dimensions maximize the enclosed area?

Visualize the chain rule applied to f(g(x)) where f(u) = u² and g(x) = sin(x).

Show that the derivative of sin(x) is cos(x) by visualizing the slope of the sine curve at each point.

A circular ripple expands at 2 m/s. How fast is the area increasing when the radius is 10 m?

A 5-meter ladder slides down a wall at 0.5 m/s. How fast is the bottom moving when the top is 3 m high?

Show concave-up and concave-down regions and inflection points of f(x) = x⁴ - 4x³.