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Unit 4
5–10% of examHere derivatives go to work in real situations. You'll explain what a derivative means with units, describe a particle moving along a line, solve related rates problems, estimate values with a tangent line, and use L'Hospital's Rule on limits that come out as 0/0 or ∞/∞.
Longer videos that cover the whole unit. Good for a first pass or a final review.
f′(a) tells you how fast the output is changing when the input is a, in output units per input unit. A full answer names the quantity, the moment and the units, for example "at t = 3 hours, the water level is rising at 2 cm per hour."
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For a particle moving along a line, velocity is the derivative of position, v(t) = x′(t), and acceleration is the derivative of velocity, a(t) = v′(t). The sign of velocity gives the direction, and speed is increasing when velocity and acceleration have the same sign.
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Derivatives describe rates in any setting: how fast a population grows, a tank drains, or cost rises with each extra item made. The meaning and units come from the quantities in the problem.
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In a related rates problem, two or more quantities change over time and are linked by an equation. Differentiating that equation with respect to time t links their rates, like dV/dt and dr/dt for a balloon being inflated.
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Draw a picture, write an equation linking the quantities (often from geometry, like the Pythagorean theorem, similar triangles or a volume formula), differentiate with respect to time, and only then plug in the values for that instant.
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Close to x = a, the tangent line L(x) = f(a) + f′(a)(x − a) is a good approximation of f(x). If the graph is concave up there, the tangent line estimate is too low (an underestimate); if it is concave down, the estimate is too high.
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If a limit of f(x)/g(x) comes out as 0/0 or ∞/∞, L'Hospital's Rule says it equals the limit of f′(x)/g′(x), as long as that new limit exists. Differentiate the top and bottom separately, not with the quotient rule, and show the 0/0 or ∞/∞ form first; the rule does not apply to other limits.
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