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241fin

241fin - Calculus I(Math 241 Final Problem 1[15 Points...

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Calculus I (Math 241) – Final Problem 1. [15 Points] Calculate the derivatives: d dx x + 1 x 2 + 1 and d dx (2 x + sin 2 x ) 3 and d dx sec( x 2 + x ) Problem 2. [10 Points] Differentiate f ( x ) = x 2 using first principles. I.e., express f 0 ( a ) as a limit of a difference quotient and work out the limit. (Make sure to show every step. This may take you 6 lines!) Problem 3. [15 Points] The point ( x, y ) = (2 , 3) lies on the hyperbola x 2 8 3 - y 2 18 = 1 . 1. Find the equation of the tangent line to the hyperbola through this point. 2. Suppose (2 . 2 , Y ) is a point on the hyperbola. Use approximation of differentials to find an approximate value for Y . Problem 4. [5 Points] Suppose f ( x ) is a function and c is an interior point of its domain. What does it mean that f ( x ) is continuous at c ? (Give a precise definition.) Problem 5. [20 Points] You are building a box. The base is square, and it has a bottom but no top. Find the side length and height of the box with the largest volume if you use A = 100 square feet of material.

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241fin - Calculus I(Math 241 Final Problem 1[15 Points...

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