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review - General Review(1 Limit and continuity one problem...

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General Review (1). Limit and continuity, one problem, emphasis will be on calculation of limit, sandwich theorem, sequences, bounds and whether a given function is continuous. (2). Derivatives, one problem, emphasis will be on rules (chain rule, product rule, quotient rule), mean value theorem and the use of L’Hospital’s rule for calculation of limit. (3). Integral, two problems, emphasis will be on Riemann sum on series, techniques of integration (substitution method, integration by parts, integral of rational functions), volume of revolution. (4). Partial derivatives, two problem, chain rule, Jacobian, extrema of function, Lagrangian multiplier. (5). Multiple integrals, two problems, double and triple integrals, transformation to curvilinear coordinates, mass and moment of inertial, special integral. (6). Matrix and vector, one problem, Cauchy-Schwarz and triangle inequalities, transpose of matrix and symmetric matrix. (7). System of linear equations, one problem, gauss elimination, existence, pivot and free variable, matrix inversion. (8). Vector space, two problems, linear independence, dimension, basis, linear span, homogeneous equation, column and null spaces, sum and intersection. (9). Linear mapping, one problem, kernel and image.
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Limits and Continuity Find limit: lim 𝑛→∞ 𝑛 + 2 − 𝑛 + 1 Solution: lim 𝑛→∞ 𝑛 + 2 − 𝑛 + 1 = lim 𝑛→∞ 𝑛+2− 𝑛+1 𝑛+2+ 𝑛 𝑛+2+ 𝑛+1 lim 𝑛→∞ 𝑛 + 2 − 𝑛 + 1 𝑛 + 2 + 𝑛 + 1 = lim 𝑛→∞ 1 𝑛 + 2 + 𝑛 + 1 = 0 Find limit: lim ?→0 0 𝑥 sin ? 3 𝑑? ? 4 Solution: This is a 0 0 indeterminate form. Applying L’Hospital rule: lim ?→0 0 ? sin ? 3 ?? ? 4 = lim ?→0 sin ? 3 4? 3 = 1 4 lim ?→0 sin ? ? = 1 4
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Limits and Continuity Find limit: lim 𝑛→∞ 1 𝑛 𝑖=1 𝑛 cos 𝑖𝜋 2𝑛 Solution: lim 𝑛→∞ 1 𝑛 𝑖=1 𝑛 cos 𝑖?
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