The City Council may employ a special engineer to make or assist the City Engineer in making any particular estimate, survey or other work. The special engineer shall make a record of the minutes of his or her surveys and all other work done for the city. He or she shall, when directed by the City Council, accurately make all plats, sections, profiles and maps as may be necessary in the judgment of the City Council. He or she shall, upon request of the City Council, make estimates of the costs of labor and material which may be done or furnished by contract with the city, and make all surveys, estimates and calculations necessary for the establishment of grades, bridges or culverts and for the building, constructing or repairing of any public improvement of the city. All records of the special engineer shall be public records which shall belong to the city and shall be turned over to his or her successor. He or she shall, when directed by the City Council, inspect all works of public improvement, and if found to be properly done, shall accept the same and report his or her acceptance to the City Council. He or she shall estimate the cost of all proposed city utilities and public improvements, together with any extensions thereof which the City Council may propose to construct or improve.