Letf(x)=√8 x+1andtakex0 =0.

1. (i)  Find the third-order Taylor polynomial p3(x) of f(x), based at x0.

2. (ii)  Use p3(x) to obtain an approximation to √8 1.08.

3. (iii)  Use the remainder term in the Taylor development to estimate the error in your approxi- mation. Explain why the first 5 digits of this approximation are exact.

Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.) 24x18x2 x f(x) - 8. Find f (x) 1+3Vx, 4)24 Need Help?Read I 9. A particle is moving with the given data. Find the position of the particle. y(t) = sin(t)-cos(t), s(0) = 2 s(t)

6. 1 points SCalcET8 3.10 025 Use a linear approximation (or differentials) to estimate the given number. (Round your answer to five decimal places.) V65 Need Help? I nesus. 1 s as nr 1 Read It Watch It to a

The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer-) crn2 What is the relative error? (Round your answer to three decimal places.) (b) Use differentials to estimate the maxim um error in the calculated volume. (Round your answer to the nearest integer.) What is the relative error? (Round your answer to three decimal places.) Need Help? Read It

Let f(x) = sin(a). Sketch the graph of f over the interval x epsilon [-pi/2, pi/2]. Use a 1-1 scale (1 unit in the horizontal is as long as 1 unit in the vertical). Estimate the slope of the line tangent to the graph at x0 = 0 as follows. Make a table whose first row contains a set of values of h (of your choice) approaching zero, and whose second row contains the slopes of the secant lines through (x0, f(x0)) and (x0 + h, f(x0 + h)). Then state the estimated limiting tangent slope. Find the equation for the tangent line to the graph of f at x = x0. Add a sketch of the tangent line and one sample secant line for one value of h in your figure in (a). Let f(x) = x3/2. Sketch the graph of f over the interval x epsilon [0, 2]. Use a 1-1 scale. Estimate the slope of the line tangent to the graph at c = 1 as follows. Make a table whose first row contains a set of special values of x (of your choice) approaching c, and whose second row contains the slopes of the secant lines through (c, f(c)) and (x, f(x)). Then state the estimated limiting tangent slope. Find the equation for the tangent line to the graph of f at x = c. Add a sketch of the tangent line and one sample secant line for one special values of x in your figure in (a). Plot a graph of the given functions and find the given limit. Show the value of the limit on the y-axis in your graph. If the limit does not exist, explain why. g(x) = 2x + x2, lim x rightarrow -2 (2x + x2) g(h) = 2 - h/2, lim x rightarrow 1 (2 - h/2) h(t) = 1 - t2 / 1 - t, lim t rightarrow 1 1 - t2 / 1 - t f(x) - 1 / x - 1, lim x rightarrow 1 1 / x - 1

/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/ 1/1/1/1/1/1ハハハハハハハハハ / 2· In this blem, we explore two functions that are not presented in a way that you may not be used to ppose f(ax) is defined on the interval (0, 1] by the following rule: we f(x) is the first digit in the decimal expansion for or instance, f(0.719)-7 and f(e-l ) = 3. But note that some numbers have two deci al expansions r instance, 1-0.5-0.499 , , and 2ō 0.35-0.34999 . . .. In such cases, one decinnal expansion s with an infinite trail of 9's and the other terminates. We shall agree to the con will always use the terminating decimal expansion. Hence f(1-5 and f(h) = 3. i. Sketch the graph of y = f(x) with appropriate scales for x and u. ii. Explain why y f(x) is integrable on [0,1]. iii. Calculate the integral f(x) dz. Can you use fundamental theorem of calculus? (b) Suppose g(x) is defined on the interval (0, 1] by the following similar rule. g(x) is the second digit in the decimal expansion forT Again we use the convention that if z has two such expansions, we use the terminating expansion.) For example g(0.719) = 1, g(e-1-6, g() = 0, and gun)-5. i. Explain why y = g(x) is integrable on [0,1]. i. Calculate the integral 10 f() dz. (A graph may help, but is not required.) Can you use the fundamental theorem of calculus? (c) Bonus! It actually turns out that it does not matter which decimal expansion we choose in the definition of either f(x) or g(x). For either convention, we get the same value of the integral. In fact, if a is a number with two decimal expansions for which the first digits differ, we can just leave f(a) undefined. (Similarly, if b is a number with two decimal expansions for which the second digits differ, we can just leave g(a) undefined.) Explain why this is true.