The slope of the tangent line at any point (x, y) on a curve is 3 √𝑥. If the point (9, 4) is on the curve, find an equation of the curve.

Answer 1

dy/dx = 3 √( x )

y( 9 ) = 4

∫_y(9)^y dy = ∫₉^x 3 √( x ) dx

y|₄^y = 2 x √( x ) |₉^x

y - 4 = 2 x √( x ) - 54

∴ y = 2 x √( x ) - 50 = 2 x^3/2 - 50

Double Check: y( 9 ) = 2 ( 9 ) √( 9 ) - 50 = 2 * 9 * 3 - 50 = 54 - 50

y( 9 ) = 4

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Answer 2

given: dy/dx = 3x^.5 thus y = 2x^1.5+C ... use point (9,4) to find C by 9=2(4)^3/2+C, C= -7. The equation of the curve is y=2x^1.5-7