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Lösung 1.1:2f

Aus Online Mathematik Brückenkurs 2

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K
K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
Zeile 1: Zeile 1:
We can rewrite the function using a trigonometric addition formula,
We can rewrite the function using a trigonometric addition formula,
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{{Displayed math||<math>f(x) = \cos\Bigl(x+\frac{\pi}{3}\Bigr) = \cos x\cdot\cos \frac{\pi}{3} - \sin x\cdot\sin\frac{\pi}{3}\,\textrm{.}</math>}}
+
{{Abgesetzte Formel||<math>f(x) = \cos\Bigl(x+\frac{\pi}{3}\Bigr) = \cos x\cdot\cos \frac{\pi}{3} - \sin x\cdot\sin\frac{\pi}{3}\,\textrm{.}</math>}}
If we now differentiate this expression, <math>\cos (\pi/3)</math> and <math>\sin (\pi/3)</math> are constants and we obtain
If we now differentiate this expression, <math>\cos (\pi/3)</math> and <math>\sin (\pi/3)</math> are constants and we obtain
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
f^{\,\prime}(x)
f^{\,\prime}(x)
&= \frac{d}{dx}\,\Bigl(\cos x\cdot\cos\frac{\pi}{3} - \sin x\cdot\sin\frac{\pi}{3} \Bigr)\\[5pt]
&= \frac{d}{dx}\,\Bigl(\cos x\cdot\cos\frac{\pi}{3} - \sin x\cdot\sin\frac{\pi}{3} \Bigr)\\[5pt]
Zeile 14: Zeile 14:
If we then use the addition formula in reverse, this gives
If we then use the addition formula in reverse, this gives
-
{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
f^{\,\prime}(x)
f^{\,\prime}(x)
&= -\Bigl(\sin x\cdot\cos\frac{\pi}{3} + \cos x\cdot\sin\frac{\pi}{3}\Bigr)\\[5pt]
&= -\Bigl(\sin x\cdot\cos\frac{\pi}{3} + \cos x\cdot\sin\frac{\pi}{3}\Bigr)\\[5pt]

Version vom 12:51, 10. Mär. 2009

We can rewrite the function using a trigonometric addition formula,

f(x)=cosx+3=cosxcos3−sinxsin3. 

If we now differentiate this expression, cos(3) and sin(3) are constants and we obtain

f(x)=ddxcosxcos3−sinxsin3=cos3ddxcosx−sin3ddxsinx=cos3(−sinx)−sin3cosx.

If we then use the addition formula in reverse, this gives

f(x)=−sinxcos3+cosxsin3=−sinx+3.


Note: In the next section, we will go through differentiation rules which make it possible to differentiate the expression directly without rewriting in this way.

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_1.1:2f“