Answer:
- general slope: y' = mtan = 4x +4
- slope at x = -2: -4
- slope at x = 0.5: +6
Explanation:
You want the slope of a line tangent to the curve y = 2x² +4x, and a graph of the tangent lines at x=-2 and x=0.5.
Slope
The slope of the curve is given by the derivative of the polynomial:
y' = 4x +4 . . . . . . general expression for slope at P(x, y)
Tangent at x=-2
The slope at x = -2 is ...
y' = 4(-2) +4 = -4
The value of y at x=-2 is ...
y = (2(-2) +4)(-2) = 0
In point-slope form, the equation of the tangent line with slope -4 at point (-2, 0) is ...
y -k = m(x -h) . . . . . . . line with slope m through point (h, k)
y -0 = -4(x +2) . . . . . line with slope -4 through point (-2, 0)
y = -4x -8 . . . . . tangent line at x = -2
Tangent at x = 0.5
The slope at x = 0.5 is ...
y' = 4(0.5) +4 = 6
The function value is ...
y = (2(0.5) +4)(0.5) = 2.5
The point-slope equation of the tangent is ...
y -2.5 = 6(x -0.5) . . . . . tangent line at x = 0.5
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Additional comment
The derivative of a function y can be signified by a prime: y'. The derivative is the slope. The slope of the curve is the same as the slope of a tangent to the curve, so we have ...
mtan = y' = 4x+4