1. tanh-1(x) is the inverse of the hyperbolic tangent function. Show that Proof. Use implicit differentiations with respect to x: Now let us express cosh y in terms of x (recall that ):

A(x) is an area bounded by l(x) and the x and y-axis. 2. Let l(x) be a line passing through the points (3,5) and (x,0), x > 3. A(x) is an area bounded by l(x) and the x and y-axis. a). Determine A(x) as a function of x. (Hint: write down an equation for line l(x) and find its y-intersect). b). Find the value of x which minimizes A(x). Justify your answer. 1 3 2 x 5 A(x) t y l(x) b a). Line l=kt+b (where k is a slope, b is y-intersect) passes through (3,5) and (x,0). Therefore A(x) is an area of the yellow triangle (see the sketch). l(x) intersects y-axis at the point (b,0) (that’s why b is called y-intersect!). Thus b). Minimize A(x). First let’s find critical points: A(x) has a minimum at x=6. A(6)=30.

3. Find the dimensions of a rectangle with area 1000 ft2 whose perimeter is as small as possible. Minimize perimeter b First let’s find critical points: Now let’s check that P(a) has a minimum at this point: Indeed, rectangle with dimensions (actually, square) has a given area and minimal perimeter.

4. b

5. The Riemann sum has a limit of the form Determine a, b and f(x). Do not evaluate the integral. There are two equivalent solutions: Case 1: or Case 2:

6. 7.