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Section 3.5 : Derivatives of Trig Functions

For problems 1 – 3 evaluate the given limit.

1. \(\displaystyle \mathop {\lim }\limits_{z \to \,0} \frac{{\sin \left( {10z} \right)}}{z}\) Solution
2. \(\displaystyle \mathop {\lim }\limits_{\alpha \to \,0} \frac{{\sin \left( {12\alpha } \right)}}{{\sin \left( {5\alpha } \right)}}\) Solution
3. \(\displaystyle \mathop {\lim }\limits_{x \to \,0} \frac{{\cos \left( {4x} \right) - 1}}{x}\) Solution

For problems 4 – 10 differentiate the given function.

1. \(f\left( x \right) = 2\cos \left( x \right) - 6\sec \left( x \right) + 3\) Solution
2. \(g\left( z \right) = 10\tan \left( z \right) - 2\cot \left( z \right)\) Solution
3. \(f\left( w \right) = \tan \left( w \right)\sec \left( w \right)\) Solution
4. \(h\left( t \right) = {t^3} - {t^2}\sin \left( t \right)\) Solution
5. \(y = 6 + 4\sqrt x \,\csc \left( x \right)\) Solution
6. \(\displaystyle R\left( t \right) = \frac{1}{{2\sin \left( t \right) - 4\cos \left( t \right)}}\) Solution
7. \(\displaystyle Z\left( v \right) = \frac{{v + \tan \left( v \right)}}{{1 + \csc \left( v \right)}}\) Solution
8. Find the tangent line to \(f\left( x \right) = \tan \left( x \right) + 9\cos \left( x \right)\) at \(x = \pi \). Solution
9. The position of an object is given by \(s\left( t \right) = 2 + 7\cos \left( t \right)\) determine all the points where the object is not moving. Solution
10. Where in the range \(\left[ { - 2,7} \right]\) is the function \(f\left( x \right) = 4\cos \left( x \right) - x\) is increasing and decreasing. Solution