Tangents on y = e^kx: the gradient is k times the height

Year 12 Mathematics Advanced · the gradient is measured from the curve, not looked up · e ≈ 2.718 · axes x from −3 to 3, y from −2 to 10

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y = e^kx tangent marks (x, m)

Drag the point along the curve, or use the x slider. Press Mark (x, m) to plot the gradient you measured.

x position 1

k in e^kx 1

k skips zero. k = 1 is the plain exponential y = e^x.

point (1, 2.72)

gradient m (measured) 2.72

m ÷ y 1.00

tangent y = 2.72x + 0.00

marks placed 0

Set k = 2, place marks, then switch both candidate curves on.

On y = e^x, move the point until the tangent gradient reads 3.0.

Rise over run across a tiny step of h = 0.0001 either side of the point, so nothing here assumes the answer.