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Solutions to exam 3 questions related to functions, calculus, and graphing for math 1100-005. It includes finding vertical and horizontal asymptotes, derivatives, integrals, and evaluating limits. The document also covers topics such as u-substitution and concavity.

Typology: Exams

Pre 2010

1 / 4

Download Math 1100-005 Exam 3 Solutions: Functions, Calculus, and Graphs and more Exams Quantitative Techniques in PDF only on Docsity! Exam 3 Review SOLUTIONS Math 1100-005 1. For the following function, find both vertical and horizontal asymptotes: (a) f(x) = 3x 2+100 4x2โ100 Vertical Asymptotes (where f(x) is undefined โ set denominator equal to zero) 0 = 4x2 โ 100 100 = 4x2 25 = x2 x = 5 AND x = โ5 are vertical asymptotes Horizontal Asymptotes: lim xโโ 3x2 + 100 4x2 โ 100 = 3 4 lim xโโโ 3x2 + 100 4x2 โ 100 = 3 4 Then the horizontal asymptote is y = 3 4 2. Find the derivative of the following functions. (a) f(x) = 1 โ ex+ 1 x + x2e2x f โฒ(x) = โex+ 1 x (1 โ 1 x2 ) + 2xe2x + 2x2e2x = โex+ 1 x (1 โ 1 x2 ) + 2x(1 + x)e2x (b) f(x) = ln ( x2โ1 x3โ3xโ1 ) First simplify f(x) f(x) = ln ( x2 โ 1 x3 โ 3x โ 1 ) = ln(x2 โ 1) โ ln(x3 โ 3x โ 1) Now, take the derivative f โฒ(x) = 2x x2 โ 1 โ 3x2 โ 3 x3 โ 3x โ 1 3. Evaluate the following integrals. (a) โซ x + 1xdx โซ x + 1 x dx = x2 2 + ln(x) + C (b) โซ e2x+5 โ 3x3 โ 1dx Need u-substitution for e2x+5 part: u = 2x + 5 du dx = 2 โ du 2 = dx Then, splitting up the integrals and substituting in u and du 2 โซ e2x+5 โ 3x3 โ 1dx = โซ e2x+5dx โ โซ 3x3dx โ โซ 1dx = โซ eu du 2 โ โซ 3x3dx โ โซ 1dx = eu 2 โ 3x4 4 โ x + C Now, substituting u = 2x + 5 back in for u โซ e2x+5 โ 3x3 โ 1dx = e2x+5 2 โ 3x4 4 โ x + C (c) โซ 3x2โ1 4x3โ4xdx Need u-substitution: u = 4x3 โ 4x du dx = 12x2 โ 4 โ du 4 = (3x2 โ 1)dx Then, substituting in u and du 4 โซ 3x โ 1 4x3 โ 4x dx = โซ 1 4u du = 1 4 ln(u) + C Now, substituting u = 4x3 โ 4x back in for u โซ 3x โ 1 4x3 โ 4x dx = 1 4 ln(4x3 โ 4x) + C