Question B1: Consider the polynomial function f(x) = 2x 14x + 13x 2. (a) Show that if |x| < 2 then |(x)| < 100. Hint: Look at each term of the polynomial, and then apply the triangle inequality. (b) Evaluate lim f(x). x-2 (c) Show that f(x) does not have a minimum value on (2,2). Question B2: Suppose that is an angle between 0 and 17. Consider the shaded region in the figure to the right. Its boundary includes the x-axis, the y-axis, the line y = sin, and the unit circle (y = 1 - x). Determine the area of the shaded region (as a function of 0). = Hint: Find the intersection between y = 1-x and y = sin 0. A line from the origin to this point splits the region into a triangle and a sector of the circle. You can calculate the area of the triangle and the area of the sector, and then add them together. Question B3: y = 1 - x y = sin (0) y = 0 Compute the following limits involving trigonometric functions, if they exist. If a particular limit does not exist, explain how you know this. sin 5x (a) lim x0 7x 6x (b) lim 1 cos 8x (c) lim Xx x40 cos 3x _ 1 (d) lim x0 sin 4x x 0 COS X - 1 (e) lim x2 x0 cos 2x 1 Question B4: The function f(x) = sin does not have a limit as x 0 (see Example 12 of section 2.1). This doesn't stop functions related to f from being well behaved. Let g(x) = 3x sin (7/7). Show that whenever 0 < |x| < we get the property -|x| < g(x) < |x|, then use the pinching theorem to evaluate lim g(x). 3 Question B5: 0 4, then x + sin x > 3. (b) Show that if x < 2, then x + sin x < 3. (c) Use the bisection method to estimate a value c so that c + sin c = 3. Work until your estimate is within 32 of the true value of c.