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Application of Derivative - 1
Meeting 7
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Tangent LineWe say that a line is tangent to a curve when the line touches or intersects the curve at exactly one point.
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Definition of Tangent Line with Slope m
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Tangent Line
If the derivative of f is m at a point (x0, y0), then the tangent line equation of is y − y0 = m(x − x0)
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Example
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Tangent Lines to the Graph of a Nonlinear Function
The slope ism= 2(-1) = -2
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Normal LineThe normal line to a curve at one of its points (x0, y0) is the line that passes through the point and is perpendicularto the tangent line at that point.
If m ≠ 0 is the slope of the tangent line, then the normal line equation is
y − y0 = −(1/m)(x − x0)
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Example
Find equations of the tangent and normal lines to the parabola at the point (−1, 4).
Ans. y + 8x + 4 = 0; 8y − x − 33 = 0
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Exercise
Find equations of the tangent and normal lines to y = f (x) = x3 − 2x2 + 4 at (2, 4).
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Finding Related Rates
The important use of the Chain Rule is to find the rates of change of two or more related variables that are changing with respect to time.
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For example, when water is drained out of a conical tank, the volume V, the radius r, and the height h of the water level are all functions of time.
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Problem Solving with Related Rates
Based on previous example, we have
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ExampleAir is being pumped into a spherical balloon (see below Figure) at a rate of 4.5 cubic feetper minute. Find the rate of change of the radius when the radius is 2 feet.
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Exercises
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Exercises