Math 41, Homework Consider the graph of function f shown below. Find the following limits or explain why they do not exist: (a) lim

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1 SSEA Summer 2016 Math 41, Homework-1 1. Consider the graph of function f shown below. Find the following limits or explain why they do not exist: (a) lim t 1 f(t). (b) lim t 0 f(t). (c) lim f(t). t 0.5 x 2. Explain why lim x 0 x does not exist. 3. Let f(x) = x2 4x + 4. What can you say about: x 3 + 5x 2 14x lim f(x) and lim f(x)? x 0 x 2 4. Prove π e < e π by referring to the following figure and choosing appropriate values for a and b. Hint: Slopes of line L a and L b.

2 SSEA Math Module Math 41 Homework-1, Page 2 of 6 August 9, 2016 y L a L b y = ln(x) a b x 5. If f(1) = 5, must lim x 1 f(x) exist? If it does, must lim x 1 f(x) = 5? Can we conclude anything about lim x 1 f(x)? Explain your answers. 6. Compute the following limits: (a) lim t 1 t 2 + 3t + 2 t 2 t 2. x 1 (b) lim. x 1 x ( ) 2 (c) lim x 4 cos. x 0 x (d) lim x 0 xe sin(π/x). 7. Which of the following statements are true, and which are false? If true, say why; if false, give a counterexample (that is, an example confirming the falsehood). (a) If lim x a f(x) exists but lim x a g(x) does not exist, then lim x a (f(x) + g(x)) does not exist. (b) If neither lim x a f(x) nor lim x a g(x) exists, then lim x a (f(x) + g(x)) does not exist. (c) If f is continuous at a, so is f. (d) If f is continuous at a, so is f. 8. If lim x 0 + f(x) = A and lim x 0 f(x) = B, find:

3 SSEA Math Module Math 41 Homework-1, Page 3 of 6 August 9, 2016 (a) lim x 0 + f(x 3 x). (b) lim x 0 f(x 3 x). (c) lim x 0 + f(x 2 x 4 ). (d) lim x 0 f(x 2 x 4 ). 9. A line y = b is a horizontal asymptote of the graph of a function y = f(x) if either: lim f(x) = b, or lim f(x) = b. x x (a) Find the horizontal asymptote of the curve y = 2 + sin(x) x. (b) Does the graph of the function f cross its horizontal asymptote at all? 10. For what value of a is: f(x) = { x 2 1, x < 3 2ax, x 3 continuous at every x? 11. A continuous function y = f(x) is known to be negative at x = 0 and positive at x = 1. Given this fact, why does the equation f(x) = 0 has at least one solution between x = 0 and x = 1? Illustrate with a sketch. Hint: Intermediate value theorem. 12. If the product function h(x) = f(x) g(x) is continuous at x = 0, must f(x) and g(x) be continuous at x = 0? Give reasons for your answer. 13. Refer to Figure (1) which shows the graphs of several functions over a closed interval D. At what domain points does each of the function appear to be: (a) Differentiable? (b) Continuous but not differentiable? (c) Neither continuous nor differentiable? Give reasons for your answers. 14. Find values of constants m and c for which the function: { sin(x), x < π, y = mx + c, x π is: (a) Continuous at x = π. (b) Differentiable at x = π.

4 SSEA Math Module Math 41 Homework-1, Page 4 of 6 August 9, 2016 Figure 1: 15. Suppose that a function f satisfies the following conditions for all real values of x and y: f(x + y) = f(x) f(y), Show that f (x) = f(x) for all values of x. f(x) = 1 + xg(x), where lim x 0 g(x) = 1. Hint: Use the limit definition of the derivative.

5 SSEA Math Module Math 41 Homework-1, Page 5 of 6 August 9, What is d9999 cos(x)? dx Use chain rule to compute the derivative of: ( y = 4 sin 1 + ) x. 18. Use the implicit rule of differentiation to compute dy/dx if: ( ) 1 y sin = 1 xy. y 19. Implicit Rule: Find the two points where the curve x 2 + xy + y 2 = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents? 20. Chain Rule: Suppose that f(x) = x 2 and g(x) = x. Then the composites: (f g)(x) = x 2 = x 2 and (g f)(x) = x 2 = x 2 are both differentiable at x = 0 even though g itself is not differentiable at x = 0. Does this contradict the chain rule? Explain? 0 0 Optional Challenge Problems. 21. What is 0 0? If we try to apply familiar rules of exponents, we get both 0 and 1 as possible answers. What value would you like 0 0 have? (a) Calculate x x for x = 10 1, 10 2, 10 3 etc., as long as possible with your calculator. What pattern do you see for the values of x x? (b) Calculate (1/x) 1/ ln(x) for x = 10 1, 10 2, 10 3 etc., as long as possible with your calculator. What pattern do you see now? Does the limit value involve a well known irrational number? 22. If lim x c (f(x) + g(x)) = 3 and lim x c (f(x) g(x)) = 1, find lim x c f(x) g(x) 23. Antipodal points: Is there any reason to believe that there is always a pair of antipodal (diametrically opposite) points on the earth s equator where the temperatures are the same? Explain. Hint: Intermediate value theorem 24. Prove that: lim (1 + x 0 x)1/x = e, by computing the derivative of f(x) = ln(x) at x = 1 using the limit definition of the derivative. That is, use: f f(1 + h) f(1) (1) = lim. h 0 h

6 SSEA Math Module Math 41 Homework-1, Page 6 of 6 August 9, The period of a clock pendulum: The period T of a clock pendulum (time for one full swing and back) is given by the formula: T 2 = 4π2 L, g where T is measured in seconds, g = 9.8 m/s 2, and L, the length of the pendulum is measured in meters. Find approximately: (a) The length of a clock pendulum whose period is T = 1 sec. (b) The change dt in T if the pendulum with a period of T = 1 sec is lengthened by 0.01 m. (c) The resulting amount the clock gains or loses in a day as a result of the change in period by the amount dt that was found in the previous part.