Michael W.SI Math 1651. Suppose that we know that f(x) is continuous and differentiable on[6, 15]. Let’s also suppose that f(6) = −2 and that f0(x) ≤ 10. Whatis the largest possible value for f(15)? (Hint: Use the Mean ValueTheorem)2. Suppose a function f(x) is continuous and differentiable everywhere.Suppose also that it has exactly two roots. Show that f0(x) has at leastone root. (Hint: Suppose the roots of f are a and b, and use the MeanValue Theorem)3. For each of the following functions, find the number in the given intervalthat satisfies the conclusion of the Mean Value Theorem(a) f(x) = x2− 5x + 7, −1 ≤ x ≤ 3(b) f(x) = x3− 6x2+ 9x + 2, 0 ≤ x ≤ 4(c) f(x) = sin(2x) + cos(x), 0 ≤ x ≤ π (calculator problem)(d) f(x) =3p(x + 1)2, −3 ≤ x ≤ 44. Find the following antiderivatives.(a)Z82xdx(b)Z(7x9+ 6x5− 42x + 0.5)dx(c)Zcos(2x)dx(d)Z(x5.4+ π)dx(e)Z76x3+113dx(f)Z√9x + 2dx(g)Z(3x2− 1)(x3− x + 1)dx15. Show that y = x−32is a solution to the differential equation4x2y00+ 12xy0+ 3y = 0 for x > 0.6. For the differential equationdydx=√xy, find the particular solutionwhen x = 1, y = 2.7. Evaluate the following(a)10Xi=1π(b)nXi=1ai= 10 andnXi=1bi= 7. FindnXi=1(3ai− 5bi)(c)nXi=1(i − 7)28. Consider the function f(x) = 2x+3 on the interval [0, 2]. Approximatethe area under the this curve by dividing the interval into 4 subintervalsof equal length, and finding the area of the four corresponding rectan-gles that result by using inscribed rectangles. This means that you usethe left-hand endpoints of each subinterval to evaluate the height ofeach
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