Question

1. (7 points) A rectangle is located with its base along the x axis, one corner at (8,0) and the opposite corner on the graph y = ln(x) for some 1 x 10. Draw a picture of the given scenario. a. If the other corner along the x axis an x value of e, what is the exact area of the rectangle? Make a table of 5 possible for the rectangles. Your table should contain columns for the x-coordinate of the other corner on the x-axis, the height of the rectangle, and the total area (exact values only). d. b. c. For what value of x is the area of this rectangle the largest? Clearly show all calculus. You may use your calculator to solve algebraic equations, but be sure to show me what you are putting into your calc. Round to three decimals if your calculator does not give exact values. (4 points) Given a differentiable function, f (x), justify rigorously that a transformation of f which either shifts, stretches, compresses, or flips the graph vertically will transform the tangent line of f at any point where a in the same way. 2. For 2 bonus points, show the same is true for any horizontal transformations. (4 points) Let y=xx. This function is neither power function nor exponential, so our standard derivative rules do not apply. By either using composition (on both sides) with a carefully chosen, helpful function and implicit differentiation OR rewriting the function in an advantageous way, find an expression for ax Your final derivative should only be a function of x. Helpful tip: It may be useful to check your answer by looking at the function and it's derivative graphically 3.

We've been using L'Hopitals as well as family of functions and modeling. stuff from chapter 4 in the 7th Edition of single variable calculus book.

Answer 1

J The picfure shows the Scenario b) The area of the tecfana le AD DC Point A is on y.nx aaa function f A ()n(x) 8-x he exact area when c) The height AD. Įnrx) The table is ghown below

The table

x -coordinate	The height	The area
2	ln(2)	ln(2)(8-2) = 6*ln(2)
3	ln(3)	5*ln(2)
4	ln(4)	4*ln(2)
5	ln(5)	3*ln(2)
6	ln(6)	2*ln(2)

e afka O C)Men Using graphical calculator to find x3.535 The csitical point A'(x)>> for 1ЯХ< 3.535 and A'(x)20 on3-535<Zs8 I0

Hence the area is maximum when x-3-53s The Value of3.535 when the area is the largest.,