Derivatives and Slopes of Tangents are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Consider the relation defined by the equation tan y = x + y for x in the open interval 0 is less than or equal to x which is less than 2pi
(a) Find dy/dx in terms of y
(b) Find the x- and y- coordinate of each point where the tangent line to the graph is vertical
(c) Find d^2y/dx^2 in terms of y

1 A box with its base in the xy-plane has its four upper vertices on the surface with equation z=48-3x^2-4y^2. What is the maximum possible volume.
2 Find the differential dw for w =ysin(x+z)
3 Find the equation of the plane tangent to z=-sin((pi)yx^2) at the point P =(1,1,0)

The equations of two ellipses are i) 4x (squared) + 9y (squared) = 36 (ii) 2x (squared) + 3y (squared) = 30. A tangent to ellipse (i) meets the ellipse (ii) at the points P and Q. Show that the tangents at P and Q to ellipse (ii) are at right angles to one another. Please show this using parametric equations.

I) Find the equation of the tangent to y=x(1-x) at x=1
ii) Find the equation of the normal to y=x(1-x) at x=1
iii) Find the equations of the tangents to y=x(1-x) that pass through (-1, 1/4)

20. The graph below displays growth of a town's population y = P(t) over the next 3 years, where t is time in months.
a. Estimate how fast the population is increasing 5 months, and 20 months from now.
b. Graph y = P'(t).
In graphs of questions 2, 4 and 6, determine which is the f(x) function and which is the derivative?

All solutions must be detailed and the final answers simplified. Show all work!
1. Differentiate the given function. Simplify your answer.
2. Differentiate the given function. Simplify your answer.
3. Differentiate the given function. Simplify your answer.
4. Find the equation of the line that is ta

Please explain the solutions so that I can understand.
7. Evaluate y = Integration-sign e^(ax+b) dx .
11. Find the area of Recordi's new exponential pool, which is bounded by y=e^x, y=0, x=0 and x=3.
12. Find dy/dx for y=x^2 e^-x . Find points where the curve has horizontal tangents.
13. Find out how many bac

Colin beleives that the number of job offers he will get depends on the
number of courses in which his grade is B+ or better. He concludes from
observation that the following figures are typical:
Number of grades of B+ or better 0 1 2 3 4
Number of job offers 1 3 4 5 6
Put these numbers into a gra