Exercises

1. Calculate the first and second derivatives of each of the following functions.

   a. \(y=10t^6-4t^3+\pi\)	b. \(u=\frac{1}{w^3}\)
   c. \(z=t^{7/5}\)	d. \(p=t\,e^{-t}\)

2. Calculate each of the following derivatives.

   a. \(\frac{d}{dx}\frac{\sin\,x}{x}\)

   b. \(\frac{d}{dx}\left(\cos\,x\right)^3\)

   c. \(\frac{d}{dt}\,\ln \sqrt{3t-9}\)

3. Calculate each of the following second derivatives.

   a. \(\frac{d^2}{d \theta^2}\,\sin\,\theta\)

   b. \(\frac{d^2}{d \theta^2}\,\cos\,\theta\)

4. Calculate each of the following derivatives.

   \(\frac{d}{dt}\,e^{2t}\sin\,t\)	\(\frac{d^2}{dt^2}\,e^{2t}\,\sin\,t\)
   \(\frac{d}{dt}\,e^{2t}\,\cos\,t\)	\(\frac{d^2}{dt^2}\,e^{2t}\cos\,t\)
   \(\frac{d}{dt}\,\ln t\,\sin\,t\)	\(\frac{d^2}{dt^2}\,\sin^2\,t\)
   \(\frac{d}{dy}\,\sin\,y\,\cos\,y\)	\(\frac{d^2}{dy^2}\,\sin\,y\,\cos\,y\)

5. Find the derivative of each of the following functions.

   \(y=\sin(2t)-3\cos\,t\)	\(z=\frac{1}{1+7t}\)
   \(y=(\sin\,3t)(\cos\,2t)\)	\(y=5^x\)
   \(z=\frac{2}{t-1}\)	\(y=x^3\sin\,2x\)

6. Find the derivative of each of the following functions.

   \(y=\cos^2t\)

   \(y=\cos{(t^2)}\)

7. Find the derivative of each of the following functions.

   \(y=\ln(x^{10}-3x)^{1/3}\)	\(y=\ln(e^x)\)
   \(y=\frac{x^2}{1+2x}\)	\(f(x)=3e^{-4x}\)
   \(y=\frac{\cos\,x+x^2}{x^3}\)	\(f(x)=\sin(x^2)\)
   \(y=\frac{\sin\,x}{\cos\,x+1}\)	\(y=y(x)\), where \(y^2+x^2=3x+7\)

8. Find all antiderivatives of each of the following functions.

   \(e^{2t}\)	\(\frac{1}{t}\)
   \(t^n\), where \(n\) is a positive integer	\(\sin\,2t\)
   \(\frac{1}{t^2}\)	\(\frac{2}{t-1}\)
   \(\cos\,t\)	\(\frac{1}{1+7t}\)

9. Given the initial value problem

   \(\frac{dy}{dt}=2-\cos(\pi y)\)   with   \(y(0)=1\),

   use Euler's Method with \(\Delta t=0.2\) to approximate \(y(1)\).
10. Find the maximum and minimum values of \(y=\sin\,x+\cos\,x\) on the interval \([0,\pi]\).