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Math 19a: Modeling and Differential Equations for the Life Sciences Calculus Review Danny Kramer Fall 2013
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Derivatives
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Point Slope Concept x y
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Solve it Out 0
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Derivative Rules
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Derivatives and Operations
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Applications Position Speed/Velocity Acceleration
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Maxima and Minima x f(x) f’(x)=0
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Some Vocabulary Continuous- no holes or jumps in the graph Differentiable- continuous graph with a derivative at each point…no “cusps” ✓ X X ✓ X X
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Sample Problem
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Integrals and Antiderivatives
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Area Concept It is area under a curve, but think of it more generally as multiplying a changing rate by the elapsed time over which the rate occurs, giving you the change in quantity that the rate is measuring. x y
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Some Notation
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Antiderivative Rules and Operations
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U substitution Replace to visualize
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Integration by Parts Opposite of product rule. Test it out! What Becomes u? Log Inverse Trig (the arcs) Algebra Trig Exponential
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Taylor Series
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Approximating Polynomial Curves x f(x) x = a f(a)
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Taylor’s Formula
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Practicing Taylor
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Parametric Curves
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Dimensions of Measurement x(t), y(t) x(y) / y(x) ? Match up x and y at any given time t. t x, y x y 5 10 x y 5 5 t0t0 tftf t0t0 tftf
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Parametric Conversion
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