0121. Solution f (a + h) f (a) (a + h)2 a2 f (a) = lim = limh0h h0 h
(a2 + 2ah + h2 ) a22ah + h2= lim = limh0hh0 h= lim (2a + h) = 2a. 2 : The DerivativeSeptember 26, 2010Derivation Notes
Let t be an increment in time and P the corresponding change
inpopulation: P = P(t + t) P(t)This depends on t, so ideally we
would want P 1 3e t+t3e tlim= lim t0 tt0 t1 + e t+t 1 + et But
rather than compute a complicated limit analytically, let
usapproximate numerically. 2 The Derivative September 26, 2010 26 / 46What
does f tell you about f ? Notes If f is a function, we can compute
the derivative f (x) at each point x where f is dierentiable, and
come up with another function, the derivative function. 0121. .
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12. 1
NotesUnderstand and state the denition of the derivative of a
function at a point. 2 The Derivative
September 26, 2010 5 / 46Outline Notes Rates of Change Tangent
Lines Velocity Population growth Marginal costs The derivative,
denedDerivatives of (some) power functionsWhat does f tell you
about f ? How can a function fail to be dierentiable? Other
notations The second derivative V63. 041, Calculus I Section 2. © 2022 SlideServe | Powered By DigitalOfficeProPost on 10-May-20151.
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2 The DerivativeSeptember 26, 2010 16 / 46Population
growth in general NotesUpshotThe instantaneous population growth is
given byP(t + t) P(t)limt0 t V63. 1 2. 10. But in see this the usual way to find derivatives is to use:On Derivative Rules it is listed as being cos(x)Done. 5 2.
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P(10)13e 9. Use the
xdenition of the derivative to ndf (2). 041, Calculus I (NYU)
Section 2. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one (or more) quantities in the problem.
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12. 12. 2 The DerivativeSeptember 26, 2010 18 / 465 6. 0121.
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12. 9601 1. 041, Calculus I Section 2. Higher Order Derivatives – In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives. You’re standing at the spot
\((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\).
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0121. ExampleDrop a ball o the roof
of the Silver Center so that its height can be described by h(t) =
50 5t 2 where t is seconds after dropping it and h is meters above
the ground. . You drop your water bottle and the water spills out. If the limit f (a + h) f (a) f
(x) f (a) f (a) = lim= limh0 h xa x a exists, the function is said
to be dierentiable at a and f (a) is thederivative of f at a.
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As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule!Implicit Differentiation – In this section we will discuss implicit differentiation. 041, Calculus I (NYU) Section 2. 016. V63. 2 The
DerivativeSeptember 26, 2010 20 / 46Marginal Cost in GeneralNotes
Upshot The incremental cost C = C (q + 1) C (q) is useful, but is
still only an average rate of change. The easiest rule in Calculus is the sum rule so make sure you understand it.
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V63. g. UpshotIf the curve is given by y = f (x),
and the point on the curve is (a, f (a)),then the slope of the
tangent line is given byf (x) f (a)mtangent = limxax a
V63. 0121. And “the derivative of” is commonly written ddx like this:ddxx2 = 2x
“The derivative of x2 equals 2x”
or simply “d dx of x2 equals 2x”It means that, for the function x2, the slope or “rate of change” at any point is 2x. 0121.
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0121. So when x=2 the slope is 2x = 4, as shown here:Or when x=5 the slope is 2x = look at here and so on. 041, Calculus I (NYU) Section 2. .