Evaluate the following limits.

• [math]\lim_{t\goesto0} \frac{\sin^2t}{t^2}[/math]
• [math]\lim_{t\goesto0} \frac{\sin t \cos t}{t}[/math]
• [math]\lim_{x\goesto0} \frac{1-\cos^2 x}{x^2}[/math]
• [math]\lim_{x\goesto0} \frac{\sin 2x}{x}[/math]
• [math]\lim_{t\goesto\pi} \frac{\sin(\pi - x)}{x(\pi - x)}[/math]
• [math]\lim_{t\goesto0} \frac{\sin2t}{\sin3t}[/math]
• [math]\lim_{x\goesto{\frac{\pi}2}} \frac{\cos x}{\left(\frac{\pi}2 - x\right)}[/math]
• [math]\lim_{x\goesto0} \frac{\cos x}{x}[/math]
• [math]\lim_{x\goesto0} \frac{1 - \cos 2x}{x^2}[/math]
• [math]\lim_{x\goesto0} \frac{1 - \cos x}{x \sin x}[/math].

Find the derivatives of the following functions.

• [math]\sin(x^2+3)[/math]
• [math]\cos e^x[/math]
• [math]\cos t \sin t[/math]
• [math]\cos^2x + \sin^2x[/math]
• [math]\cos(\sin x)[/math]
• [math]\ln \sin x[/math]
• [math]\sin^5x^5[/math]
• [math]\frac{\sin x}{\cos x}[/math]
• [math]\frac{\cos x}{\sin x}[/math]
• [math]e^{-x} \sin x[/math].

Evaluate the following integrals.

• [math]\int \cos 7x \; dx[/math]
• [math]\int (\cos 2x + \sin 3x) \; dx[/math]
• [math]\int e^x \cos e^x \; dx[/math]
• [math]\int \sin(x+a) \; dx[/math]
• [math]\int (\cos x)e^{-\sin x} \; dx[/math]
• [math]\int (\cos t)\cos(\sin t) \; dt[/math]
• [math]\int \frac{\sin x}{\cos x} \; dx[/math]
• [math]\int \cos^6x \sin x \; dx[/math]
• [math]\int \sin^6x \cos x \; dx[/math]
• [math]\int (\cos^2 x + \sin^2 x) \; dx[/math]

Find the integrals

• lab{6.2.4a} [math]\int \cos^n x \sin x \; dx[/math]
• lab{6.2.4b} [math]\int \sin^n x \cos x \; dx[/math]. The next two integrals can be reduced to sums of integrals of the forms \ref{ex6.2.4a} and \ref{ex6.2.4b} by using the identity [math]\sin^2 x + \cos^2 x = 1[/math].
• [math]\int \sin^3 x \; dx[/math]
• [math]\int \cos^4 x \sin^3 x \; dx[/math].

Express [math]\cos^2 x[/math] in terms of [math]\cos 2x[/math], and thence evaluate [math]\int \cos^2 x \; dx[/math].

Express [math]\sin^2 x[/math] in terms of [math]\cos 2x[/math], and thence evaluate [math]\int \sin^2 x \; dx[/math].

Solve the differential equations

• [math]\dydx = e^x \sin e^x[/math]
• [math]\dydx = \frac{\sin x}{\cos y}[/math]
• [math]y \dydx = \frac{\cos x}{\sin (y^2)}[/math].

Evaluate the following limits using L'H\^opital's Rule.

• [math]\lim_{x\goesto0} \frac{\sin x}{x}[/math]
• [math]\lim_{x\goesto0} \frac{1 - \cos x}{x}[/math]
• [math]\lim_{x\goesto0} \frac{\sin^2 x}{x^2}[/math]
• [math]\lim_{t\goesto0} \frac{e^t - 1}{t}[/math]
• [math]\lim_{x\goesto0} \frac{e^x - 1 - x}{x^2}[/math]
• [math]\lim_{x\goesto0} \frac{1 - \cos x}{x \sin x}[/math].

Draw the graphs of the equations

• [math]y = 3 \sin x[/math]
• [math]y = \cos \left( \frac{\pi}2 x \right)[/math]
• [math]y = \sin(2\pi x)[/math]
• [math]y = 2 \sin \left( x + \frac{\pi}6 \right)[/math]