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SLOPE FIELDS. A slope field is a graphical representation of the slope of a function at any point on the coordinate plane. We have seen how to find the slope of any function – take its derivative.
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SLOPE FIELDS • A slope field is a graphical representation of the slope of a function at any point on the coordinate plane. • We have seen how to find the slope of any function – take its derivative. • For instance, dy/dx = y. For this function, the slope at any point is equal to its y coordinate so when y = 1, dy/dx = 1; y = 2, dy/dx = 2; y = 3, dy/dx = 3; etc. • We can plot these on a coordinate plane
SLOPE FIELD OF dy/dx = y • The slope is the same for every value of y regardless of what x is. • This graphical representation gives you a visual of all the possible functions that have this slope. • The function will have the same slope at each y
SLOPE FIELD OF dy/dx = 1/x • In this case, the slope changes with x only so it is the same for every value of y. • Look at the point (1, 0.5) • We can sketch what we think the function will look like as it goes through this point. • If we take the anti-derivative of 1/x we get: • y = ln(x) + C • .5 = ln(1) + C C = .5 • y = ln(x) + .5 • Graph this and see if it matches our sketch.
EXAMPLE • Match the slope fields at right. • y’ = cos x • dy/dx = 2x • dy/dx = 3x2 – 3 • y’ = -π/2
