1. (1 point) Find all numbers c that satisfy the conclusion of Rolle's The- orem for the following function and interval. Enter the values in increasing order and enter N in any blanks you don't need to use x) : lx\\/x+ 2, [2,0] 2. (1 point) Let x) : 312 | 3.1: 10. Answer the following questions. 1. Find the average slope of the function f on the interval [1,1]. Average Slope: = 2. Verify the Mean Value Theorem by nding a number c in (1,1) such thatf'(c) z. Answer. c = 3. (1 point) A function x) and interval [(1,1)] are given. Check if the Mean Value Theorem can be appliedto fon [a,b]. If so, find all values I: in [1.1, b] guaranteed by the Mean Value Theorem Note, if the Mean Value Theorem does not apply, enter DNE for the 8 value. f(x)=131nx3 on [1,20] c = (Separate multiple answers by commas.) 4. (1 point) Suppose f is a differentiable function such that f'(x} 5 4 for all x E [1,4]. If f(1)= 1, the Mean Value Theorem says that f [4) 5 V for what value of V? (Choose V as small as possible.) V: 5. (1 point) Find the Taylor polynomial for the function f(x) = \\/ lJcJc2 about the point a = S that has three terms. (Your answers should include the variable it when appropriate.) lxxzw + + 7. (1 point) Find the degree 3 Taylor polynomial 73(x) of the function f(x) = (3x+21)4/3 about x = 2. T3 (X) =10. (1 point) According to Poiseuille's Law, the volume of blood per unit time that ows past a given position in a blood vessel is proportional to the fourth power of the radius of the vessel. In other words, F 2 ha\12. (1 point) Suppose that the Taylor polynomials of f(x) and g(x) are given by f(x) ~ 5+5x+ 5x2+5x3 and g(x) ~ 6+2x+2x2+4x3. Find the Taylor polynomial of degree 3 for the product h (x) = f(x) . 8(x) ~co+cix+czx-+czx3. CO= G1=.13. (1 point) Find the linear approximation of x) = 1111 at x = 1 and use it to estimale h1[1.28). L(x) = _ 1111.28 m _ 14. (1 point) Find the linear approximation of the function f(x) 2 #3 +1 at ID = '22, and use it to approximate V249 and \\125.1. (a) f(x) = [3 +1 m (b) m m (c) 25.1 m For parts (b) and (c), you should enter your answer as a frac- tion. If you enter a decimal, make sure that it is correct to at least six decimal places. 15. (1 point) 25 ft. The length L of a long wall is to be approximated. The angle 9, as shown in the diagram (not to scale), is measured to be 47, accurate to within 0.1. Assume that the triangle formed is a right triangle. a) What is the measured length of the wall? feet b) Estimate an upper bound for the propagated error using a linear approximation. feet 0) What is an upper bound for the percent error? 16. (1 point) The acceleration due to gravity, g, is given by GM where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant. (a) Suppose that we increase from our distance from the cen- ter of the Earth by a distance Ar = x. Use a linear approximation to find an approximation to the resulting change in g, as a frac- tion of the original acceleration: Ag ~ gx ( Your answer will be a function of x and r.) (b) Is this change positive or negative? Ag is [?/positiveegative] (Think about what this tells you about the acceleration due to gravity. ) (c) What is the percentage change in g when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.35 km; assume the radius of the Earth is 6400 km)? percent change =