4. Let f (x ) = a. Find the equation of the tangent line of the function f at the point (4. f(4)). b. Use differentials to estimate the value of f (4.02). (6 points) 5. Use implicit differentiation to prove that the curve x' + y = (2x + 2y' - x) (see exercise 27 of page 187 of the textbook) has a vertical tangent line at the point (1,0)- Mathematics 265: Introduction to Calculus I (Revision 15) (8 points) 6. One side of a right triangle is known to be 25 cm exactly. The angle opposite to this side is measured to be 60", with a possible error of 10.5. a. Use differentials to estimate the errors in the adjacent side and the hypothenuse. b. Estimate the percentage errors in the adjacent side and hypothenuse. (9 points) 7. An Earth-observing satellite can see only a portion of the Earth's surface. The satellite has horizon sensors that can direct the angle e shown in the accompanying figure. Let / be the radius of the Earth (assumed spherical) and h the distance of the satellite from the Earth's surface. Of Satellite h Earth a. Show that h = r(csce -1). b. Using / = 6378 km, find the average rate of change of the distance from the satellite to the surface of the Earth, when o changes from */4 to #/3. What are the units? c. At what rate is the distance h changing when 0 = x/6