l. (1 point) Suppose the function f is continuous on the closed interval [1,2], f(1) = 5, and f(2) = 9. Then, from the Intermediate Value Theorem we can conclude that...(select all that apply) o A. f(1.5) = 7. o B. for any value of y between 5 and 9, there exists at least one number c between 1 and 2 such that c) = y. o C. there must exist at least one number c between 1 and 2 such that f(c) = 10. o D. there must exist at least one number c between 1 and 2 such that f(c) = 7.5. 2. (1 point) Carry out three steps of the Bisection Method for f(x) = 3Jr x7 as follows: (a) Show that f(x) has a zero in [1,2]. (b) Determine which subinterval, [1, 1.5] or [15,2], contains a zero. (c) Determine which interval, [1,1.25], [1,25,15], [1.5,1.75], or [135,2], contains a zero. In part (b), the interval with a zero is In part (c), the interval with a zero is 3. (1 point) (a) Find the slope of the tangent line to the parabola y = J:2 + 3.1: at the point (3,0). (b) Find the equation of the tangent line in part (a). You must use the denition of the derivative to solve this prob- lem (a). (b)y= 4. (1 point) Match the graph of each function in A through D with the graph of its derivative 1 through 4 below. A B . i J- . C n 1. 2. _3. 4. Note: You can click on the graphs to enlarge the images. 5. (1 point) (a) Fmd the slope ofthe tangent line to the curve y :19 2x2 + 1 at the point where x = a. (b,c) Find the equations of the tangent lines at the points (b) (1.0) and (c) (2.1). You must use the denition of the derivative to solve this prob- lem (a) (b) y = (a) y = 6. (1 point) Dierentiate the following function: 39:) = g R11) = 7. (1 point) Dierentiate the following function: _ x2 + 71 8 3' ' y = s. (1 point) Find the equation of the tangent line to the curve y = x x at the point {16, 64). y = 9. (1 point) For what values of a and b is the line 2x+y = b tangent totheparabolay: as:2 whenx= 3'? a=_ b: 10. (1 point) Find the x-cooidinates of the points on the curvey = 21:3 +15):2 lr 1 where the tangent is horizontal. List these x-coordinates below. If there are multiple values, separate them with commas. x: 11. (1 point) Find the parabola with equation 3; = m2 + bx whose tangent line at (1,2) has equation 3; = 412. a=_b=_ 12. (1 point) x) = 1sz: Find the derivative of the function above using the denition of the derivative. Which of the following is the domain of the function? 0 A. Positive real numbers 0 B. All real numbers except % o C. Nonnegative real numbers 0 D. All real numbers except :3 o B. All real numbers except 0 o F. Integers o G. x = 2 Which of the following is the domain of the derivative? 0 A. Nonnegative real numbers 0 B. All real numbers except % o C. Integers o D. All real numbers except 0 o E. Positive real numbers 0 F. x = 2 o G. All real numbers except :3 13. (1 point) 12 ifxgil Let = x) {m+b ifx :> 1 Find the values of m and b that make x) differentiable. m=_ b: 14. (1 point) Consider the following eqaation: . fia+h) f(a} f'(0)=}|l_1}l1] h Letf(x)=\\3/i If a g 0, use the above formula to nd f"(a) = Hint: use the formula (11 b)(a2 +ab+b2) = a3 3:3 Now show that f"{0) does not exist and that f has a vertical tan- gent line at (0,0) x+10x 7' Note that you can do this two ways; either by simplifying rst, or by using the Quotient Rule. Both approaches should obtain the same answer. 15. (1 point) Find the derivative of F{x) = my = 16. (1 point) Differentiate: F'(y) = - 9 F (y) = ( * - 7x 3 )