1. 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 ):

2. 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. 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. 4. b

5. 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. 6. 7.