Final Exam Review Sheet - 18.01 Calculus Final Exam at...

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18.01 Calculus Jason Starr Final Exam at 9:00am sharp Fall 2005 Tuesday, December 20, 2005 More 18.01 Final Practice Problems Here are some further practice problems with solutions for the 18.01 Final Exam. Many of these problems are more difficult than problems on the exam. Goal 1. Differentiation. x 1.1 Find the equation of every tangent line to the curve y = e containing the point ( 1 , 0). This is not a point on the curve. 1.2 Let a and b be positive real numbers. Find the equation of every tangent line to the ellipse with implicit equation, 2 2 x y + = 1 , a 2 b 2 containing the point (2 a, 2 b ). This is not a point on the ellipse. 1.3 Let a be a real number different from 0. Use the definition of the derivative as a limit of difference quotients to find the derivative to the following function, 1 f ( x ) = , x at the point x = a . 1.4 Use the definition of the derivative as a limit of difference quotients to find the derivative of the following function, f ( x ) = tan( x ) , at the point x = 0. You may use without proof that the following limits exist and have the given values, sin ( x ) 1 cos( x ) lim = 1 , lim = 0 . x 0 x x 0 x 1.5 For x > 0, let f ( x ) be the function, x f ( x ) = e . Thus the inverse function, y = f 1 ( x ) , 1
18.01 Calculus Jason Starr Final Exam at 9:00am sharp Fall 2005 Tuesday, December 20, 2005 satisfies the equations, e y = x, and y = ln( x ) . Compute the derivative, dy . dx Goal 2. Sketching graphs. 2.1 Sketch the graph of the function, 1 2 1 f ( x ) = + . x 1 x x + 1 2.2 Sketch the implicit function, y 2 xy x 2 = 1 . 2.3 Sketch the graph of the function, x 2 x 2 f ( x ) = + . x + 1 x 1 Goal 3. Applications of differentiation. 3.1 A sculpture has the form of a right triangle. The material used for the vertical leg has twice the cost of the material used for the horizontal leg. The length of the hypotenuse is fixed (thus its cost is irrelevant). What ratio of vertical leg to horizontal leg minimizes the total cost of the material? 3.2 A farmer has a fence running diagonally across her property at a 45 degree angle to the north- south and east-west lines. She decides to build a corral by adding a length b a of fence running north-south, a length b a of fence running east-west, and then connect the two corners with 2 length b of fence running north-south and east-west. Thus, the total new length of fence needed is 4 b 2 a , and the corral has the form of a square of length b , with a small isosceles triangle of leg length a removed from one corner (where the square corral meets the pre-existing diagonal fence). What ratio of a to b gives maximal area of the corral for a fixed length of new fence? 3.3 An icicle has the shape of a right circular cone whose ratio of length to base radius is 10. Assuming the icicle melts at a rate of 1 cubic centimeter per hour, how fast is the length of the icicle decreasing when it is 10 centimeters long?

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