All answers have to be exact, not using decimals. Bare answers will count as "no answer": you have to show all your work, including intermediate steps, and explain how you reached your conclusion. Please write the words clearly

1. Consider the function f(x) =x³+x+ 1

You can see from its graph that it is invertible. Prove it by computing the derivative and explain why the result proves that f is invertible (it actually proves more: not only does f¹ exist, it is also differentiable).

2. A problem of considerable importance in astronomy, arising in the study of planetary motion, is the determination of the "eccentric anomaly" E of the planet. It satisfies the equation: E c sin(E) = T2t, where t is time in years, T is the period of the orbit and c is a constant called the eccentricity. Suppose c= 0.9, and T= 10 years.

a. Find E(t) when E=3

b. When is E=3?

3. During an adiabatic process in chemistry, the pressure P and volume V of a certain element in a container are always related by the equation

PV^3/2=32, where V is measured in m³(cubic meters) and P is measured in N/m²(Newtons per square meter).

a. Use differentials (or a linear approximation) to estimate the pressure when the volume is 1.1m³. Notice that when the volume is 1m, you can compute the pressure exactly.

b. Suppose that your measurement of the volume is not exact, but is off by a relative error of 2%. Find the corresponding relative error (as a percentage) for the pressure. Give the correct units.

!!!! Note If we denote the uncertainty in a measure of quantity x as x, the relative error is xx, which we approximate by linear approximation with xdx, which involves applying differentiation rules, as needed, when x is expressed as a function of another quantity, to find an expression for dx.

4.

Here is the picture of the graph of a distance function y = f (3") Distance 3; is in feet1 time :1: is in seconds 1. f'(7) = 2. Imam.m m: + 20) = f m+2n _ 3.1imm_,n m _ 4. liming fgm+2n)2n _ m2 _ 5. The average velocity on the time interval [0, 40] = . The maximum velocity on the time interval [0, 40] = 6 7. 11mm_,m f"(;r;) = 8- Let 90v) = T and Mm) = me. Find mm =