# Evaluate analytically

Evaluate analytically

Refer to the graph of the function  as shown below for the questions 3-4

Refer to the graph of the function  as shown below for the questions 5-6

7. Find an equation of the tangent line to the graph of   at  .

9. Find the intervals on which the function increases or decreases.

10. Find the intervals on which the function is concave up or is concave down and write the exact coordinates of the points of inflection.

11. Find the radius and the height of the right circular cylinder of the largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches.

Volume of a cylinder

12. A hot air balloon is rising vertically from a level field from a point that is 500 feet from an observer. At the instant, when the angle of elevation is , the angle of elevation is increasing at a rate of 0.08 radians per minute. How fast is the balloon rising at that moment?

13. Use the Newton’s method to find the solution of the equation with 3 as the starting guess, show at least five values of the iterations.

14. A power company needs to lay a cable from point A on one bank of an 800-foot wide straight river to point B on the opposite bank 1600 feet downstream. The power company will choose a point C on the same side of the river as B, and will lay a cable underwater from A to C and on land from C to B. It costs \$3 a foot to lay the cable on land and \$5 a foot to lay it underwater.

Where the point C should to be chosen to minimize the total cost and what is the minimum cost?
You may use

15. Evaluate

17.  Find the exact value of

Write the locations of the local extrema of  in the interval [0,6]

Write the location of the absolute minimum of this function in [0,6]

Write an approximation of to the nearest integer.