4.4: Summary
- Page ID
- 121103
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- The derivative is a linear operation, which means that the derivative can be distributed over a sum of functions or exchanged with a constant multiplying a function. The derivative of a polynomial is thus a sum of derivatives of power functions.
- A "second derivatives" is the derivate of a derivative.
- Antiderivatives reverse the process of differentiation. Antidifferentiation is, however, not unique. Given a function \(f(x)\), we can only determine its antiderivative up to some arbitrary constant.
- The product, quotient and chain rules are,
\[\begin{gathered} {[f(x) g(x)]^{\prime}=f^{\prime}(x) g(x)+g^{\prime}(x) f(x),} \\ {\left[\frac{f(x)}{g(x)}\right]^{\prime}=\frac{f^{\prime}(x) g(x)-g^{\prime}(x) f(x)}{[g(x)]^{2}}} \\ {[f(g(x))]^{\prime}=f^{\prime}(g(x)) \cdot g^{\prime}(x) .} \end{gathered} \nonumber \]
These allow us to compute derivatives of functions made up of simpler components for which we have already established differentiation rules.
- Given the graph of a function, we can use qualitative features - signs, zeros, peaks, valleys, and tangent lines slopes to sketch both its derivatives and antiderivatives.
- The applications examined in this chapter included:
- energy loss and Earth’s temperature; and
- position, velocity and acceleration of an object.
In particular, these are related via differentiation and antidifferentiation. Given an object’s position \(y(t)\), its velocity is \(v(t)=y^{\prime}(t)\) and its acceleration is \(a(t)=v^{\prime}(t)=y^{\prime \prime}(t)\).
- Which of the following operations are linear? (a) division (b) exponentiation (c) composition (d) squaring
- Why are the product and chain rules useful?
- Suppose an object has acceleration \(a(t)=10 \mathrm{~m} / \mathrm{s}^{2}\). What can you say about its: (a) accelleration at \(t=5 \mathrm{~s}\) ? (d) position at time \(t ?\) (b) velocity at time \(t\) ? (e) position at time \(t=5 \mathrm{~s}\) ? (c) velocity at time \(t=5 \mathrm{~s}\) ?
- Consider the following graph which describes the position \(y\) of an object at time \(t\) :
Where is the object’s velocity minimal? Maximal?