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Unformatted text preview: Differential Calculus Fall 2010 Take-Home Quiz 2 1. Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? 1) f (x) = 2) f (x) = x2 - 2x + 1 x2 - 1 x2 - 2x x2 - 5x + 6 2. Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? -x + 5, x < 1 f (x) = 2x, x1 Find the constant a such that the function is continuous on the entire real line. 3. f (x) = 9 sin x , x < 0 x a - x2 , x 0 2x + 15, x < 3 a - 4x, x 3 4. f (x) = 5. Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f (x) = x2 - 6x + 5, [2, 6], f (c) = 0 6. Find the vertical asymptotes (if any) of the function. 1) f (x) = 2) f (x) = 3) f (x) = x2 x2 - 9 x2 - 2x x2 - 4 x2 - 4 x2 - 6x + 8 7. Determine whether the function has a vertical asymptote or a removable discontinuity at x = -2. 1) f (x) = x2 - 4 x+2 2) f (x) = x2 - 4x + 4 x+2 8. Find the limit. x2 + 1 1) lim x1+ x - 1 2) lim -
x2 x2 + 1 x-2 9. Find the defivative by the limit process f (x) = x2 + 3x + 4 Use alternative form of the derivative to find the derivative at x = c (if it exists). 10. f (x) = |x + 3|, c = -3 11. f (x) = x + 1, c = -1 12. Find the derivative of the function 2 1) f (x) = + 4 cos x 3 x 2) f (x) = 3 x - 4 cos x 3) f (x) = 3x2 - 3x + 1 3 x 13. Find an equation of the tangent line to the graph of f at the indicated point 2 f (x) = 4 x3 (1, 2) 14. Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line y = x4 - 8x2 + 2 15. Find k such that the line is tangent to the graph of the function f (x) = x2 - kx, y = 4x - 9 16. When a guitar string is plucked, it vibrates with a frequency of F = 200 T , where F is measured in vibrations per second and the tension T is measured in pounds. Find the rate of change of F when 1. T = 4 2. T = 9 ...
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- Spring '11
- Calculus, Mathematical analysis, Continuous function