Compute the following limits, if they exist. Else, argue why the limit does not exist.

4

7

7

(a) Show that the equation x⁶ − 5x − 5 = 0 has at least one solution on the interval

[−1,0].

(b) Compute the derivative of directly from its definition as the limit of a

difference quotient.

Consider the function.

What is the domain of f? Find the horizontal and vertical asymptotes, local minima, local maxima, and reflection points of f. Identify the regions where the graph of f is concave up or concave down. Finally, sketch the graph.

Solve the following:

(a) Integrate

R ⁻³e^−1/x2dx

(b) Integrate x

(c) Integrate

(d) Differentiate f(t) = t^t3

Choose only two of the below:

(a) Find the area between the curves x = y² and 0 = −x − y² + 2 (in absolute terms).

(b) A farmer owns an 8km long stretch of land between two parallel rivers that are 1500m apart. What is the area of the largest rectangular enclosure he can fence off with (i) 1km of fencing and (ii) 4km of fencing, assuming that no fence is needed along the rivers?

(c) Use implicit differentiation to find an equation for the tangent line to the graph of

sin(2x + y) = y³ sin(x) at the point (0,0).