Consider the following function. f(x) = e4x2,    a = 0,    n = 3,    0 ≤ x ≤ 0.2 (a) Approximate f by a Taylor polynomial with degree n at the number a. T3(x) = ______       (b) Use Taylor's Inequality to estimate the accuracy of the approximation  f(x) ≈ Tn(x)  when x lies in the given interval. (Round your answer to five decimal places.) |R3(x)| ≤ ______

Are the following functions monotone in x? Explain your answer. f(x) = x² f(x) = √√x - x, where x € [x, +∞] and a is a positive constant. f(x, y) = 5xy² (a) (b) (c)

Use the secant method to approximate one of the real roots of f(x) and complete Table III. Use initial values at x−1=1 and x0=1.3. Perform as many iterations as necessary until ea<5%.

Consider the following equation defining the function f on the integers implicitly: f(n) = n+ f(n – 1), where f(1) = 1. Show that f(n) = n(n + 1).

Graph f(x) = 3x2/3 − x. (The 2/3 is an exponent on the x.) a. Compute the first derivative. Make a sign chart (as demonstrated in lecture) toindicate the intervals of increase and decrease. Find any local maximum andminimum values.b. Compute the second derivative. Make a sign chart (as demonstrated in lecture) toindicate the intervals of concavity. Find any inflection points.c. Sketch a neat and accurate graph using your sign charts from parts a. and b. Usegraph paper or a ruler to make axes. Be sure to mark the scale on the axes.Neatness and accuracy count for points!

1) Use Bisection to find a root of the function f(x) = ln x-2 on the interval [7, 8] to within 1 decimal place. Organize the results of all the calculations in the table below. You don't need to write down more than 5 decimal places. f(a.) C₁ ƒ (c₁) b₁ f(b) a₁ Half Interval Length

□ 5. Find(x) for the following function: f(x) = x² +4 O y=√x-4 O x= √√y-4 O ƒ-1(x)=√x -4 O ‹ −¹(x) = √√x-4

Find the smallest positive x for which the function f(x) = csc(0.79x-8.11) has a local maximum.  Provide four decimal places.

The function below has two roots between x = -6 and x = 10. Use the bisection method to estimate the lesser root and the Newton-Raphson Method for the largest root, correct up to seventh decimal placesf(x) = (x + 5)^3 - e^x

Find the Quadratic approximation for the following function at x = 3 f(x) = e1-8x