All answers have to be exact, not using decimals. And show all your work, including intermediate steps, and explain how you reached your conclusion.

!!! Please write the words clearly !!!

Thank you

1. Use the definition, not differentiation rules, to compute dxd?(x+21?)at points ?5, ?3, ?1.2

2. 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 f^?1 exist, it is also differentiable).

3. Derive ?5e^{?8 sin(?7x}6⁾

4. Let f(x) = 6x³ln(4x). Write the first three derivatives of f(x).

5. 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) = T2?t?, 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???

6. 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 x?x?, 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.

7.

A highway patrol officer's radar unit is parked behind a bulletin board 200 ft from a long straight stretch of highway US5. Down the highway, 200 feet from the point on the highway closest to the officer, is an emergency call box. A truck passes the call box and, at that moment, the radar unit indicates that the distance between the officer and the truck is increasing at a rate of 45 miles per hour. The posted speed limit is 55 miles per hour. Calculate the speed of the truck. Should the officer apprehend the driver of the truck for speeding? 200 ft. Truck 200 ft. OfficerHere 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 =