![]() ![]() ![]() Please e-mail any correspondence to Duane Koubaīy clicking on the following address About this document. Your comments and suggestions are welcome. Ĭlick HERE to see a detailed solution to problem 21.Ĭlick HERE to return to the original list of various types of calculus problems. With tangent lines parallel to the line y + x = 12. PROBLEM 21 : Find all points ( x, y) on the graph of.PROBLEM 20 : Find an equation of the line perpendicular to the graph ofĬlick HERE to see a detailed solution to problem 20.PROBLEM 19 : Find an equation of the line tangent to the graph ofĬlick HERE to see a detailed solution to problem 19.For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 18. Compare it with the ordinary product rule to see the similarities and differences.Ĭlick HERE to see a detailed solution to problem 16.Ĭlick HERE to see a detailed solution to problem 17. For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 15. For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 14. For what values of x is f'( x) = 0 ?Ĭlick HERE to see a detailed solution to problem 13. The following problems require use of the chain rule.Ĭlick HERE to see a detailed solution to problem 7.Ĭlick HERE to see a detailed solution to problem 8.Ĭlick HERE to see a detailed solution to problem 9.Ĭlick HERE to see a detailed solution to problem 10.Ĭlick HERE to see a detailed solution to problem 11.Ĭlick HERE to see a detailed solution to problem 12. ![]() In most cases, final answers to the following problems are given in the most simplified form.Ĭlick HERE to see a detailed solution to problem 1.Ĭlick HERE to see a detailed solution to problem 2.Ĭlick HERE to see a detailed solution to problem 3.Ĭlick HERE to see a detailed solution to problem 4.Ĭlick HERE to see a detailed solution to problem 5.Ĭlick HERE to see a detailed solution to problem 6. In the list of problems which follows, most problems are average and a few are somewhat challenging. Each time, differentiate a different function in the product and add the two terms together. The rule follows from the limit definition of derivative and is given by Substitute the value of r 165 m to the obtained equation. Substitute and solve for the growing rate of the oil spot. dA/dt d/dt (r 2 )2r (dr/dt) It is given dr/dt 1.2 meters/minute. The notes contain the usual topics that are taught in those courses as well as a few extra topics that I decided to include just because I wanted to. Differentiate both sides of the area equation using the chain rule. The product rule is a formal rule for differentiating problems where one function is multiplied by another. Contained in this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, II and III) as well as Differential Equations at Lamar University. In the following discussion and solutions the derivative of a function h( x) will be denoted by or h'( x). The following problems require the use of the product rule. ![]()
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