Interactive Calculus Lab for AP Physics C

Secant → Tangent Packet A2

Penguin Percy's position is graphed below. Shrink Δt and watch the secant line (orange) turn into the tangent line (dark). Its slope settles down to the instantaneous velocity.

x(t)secant linetangent line

Controls

Position function

Starting time t₀

Δt

show tangent

Readout

Average velocity Δx/Δt

Instantaneous dx/dt at t₀

Difference

Your table

Δt (s)	Δx/Δt (m/s)

Record a row, shrink Δt, record again. What number are you closing in on?

x → v → a Packet A6–A7

Drag the time slider (or drag on any graph). The tangent line on each graph has a slope equal to the value on the graph below it. In Predict mode the bottom graphs are hidden — your scrubbing leaves a trail of slope dots. Sketch the curve in your head, then reveal.

Controls

Position function

Time t

Mode

At this instant

x(t)

v(t) = dx/dt

a(t) = dv/dt

The Area Machine Packet B1, B4

Add up thin rectangles under the curve. Area above the axis counts positive (blue); below counts negative (orange). Crank up the number of rectangles and compare with the exact answer from the Fundamental Theorem of Calculus.

Controls

Function

From a =

To b =

Number of rectangles n

Rectangle height from

Result

Sum of rectangles

Exact: F(b) − F(a)

Error

Rule Drill Packet A3–A5, B2, B5

Type your answer using ^ for powers, * or nothing for multiplication, and functions like sin(2x), e^(3x), ln(x), sqrt(x). For integrals, leave off the + C. Answers are checked numerically, so any correct form counts.

I read that as: —

Problem set

Score

Correct0

Attempted0

Current streak0

Professor Waddles is ready when you are.

Drag Lab: Hank's Parachute Packet C2 · Topic 2.9

Hank the Hamster falls from rest with air resistance. The smooth curve is the exact solution from separation of variables. The dots are a step-by-step simulation that only uses Newton's second law: a = F_net/m, then v_new = v + aΔt. Shrink the step and the dots land on the curve.

exact solutionstep-by-step simulationterminal velocity

Controls

Drag model

Mass m

Drag constant b

Simulation step Δt

Results

Terminal velocity v_T

Time constant τ = m/b

Speed at t = 2 s (exact)

The math with your numbers

Slice-a-Rod Packet C1 · Topics 2.1, 5.4

A 1.0 m rod is cut into N slices. Each slice has mass dm = λ(x) dx. Add up the slices to get the mass, the center of mass, and the rotational inertia about the axis you choose. More slices → the sum becomes the integral.

Sum of slices vs. the integral

Quantity	Integral	Sum with N slices	Exact

Controls

Density λ(x) in kg/m

Number of slices N

Rotation axis at x =

Hank says: heavier slices are darker. Watch where the center of mass (orange triangle) sits when one end is denser. Then put the axis at the CM — that's the smallest I can get.