| Skill | Example Problem | |-------|-----------------| | Compute limits using ε‑δ | Prove (\displaystyle \lim_x\to2\fracx^2-4x-2=4). | | Differentiate composite functions | Find (\displaystyle \fracddx\Big(e^\sin(x^2)\Big)). | | Apply the Mean Value Theorem | Show that for (f(x)=x^3-3x) on ([1,3]) there exists (c) with (f'(c)=\fracf(3)-f(1)2). | | Evaluate definite integrals via substitution | (\displaystyle \int_0^\pi/4\tan x,dx). | | Set up and compute volumes of revolution (washer & shell) | Volume of the solid obtained by rotating (y = \sqrtx) about the x‑axis from (x=0) to (x=4). | | Expand functions in a Taylor series | Find the Maclaurin series for (\ln(1+x)) up to (x^5). | | Work with parametric curves | Compute the arc length of (x=t^2,; y=t^3) for (0\le t\le1). | | Solve a basic differential equation | Solve (\displaystyle \fracdydx=y\cos x) with (y(0)=2). |
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