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MIT 18 01 - Final Examination

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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 thes e problems are more difficult than problems on the exam. Goal 1. Differentiation. x1.1 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 2x y+ = 1, a2 b2 containing the point (2a, 2b). 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, √xf (x) = e . Thus the inverse function, y = f −1(x), 118.01 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, x2 x2 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 4b − 2a, 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? 3.4 A cube of ice rests on the ground. The cube of ice melts at a rate proportional to the surface area of the cube exposed to the air (thus, the area of the 5 sides other than the side sitting on the ground). Assuming it takes 5 hours before the volume of the melted cube equals 1/2 the initial volume, how much longer does it take for the cube to melt entirely? Goal 4. Integration. 2� 18.01 Calculus Jason Starr Final Exam at 9:00am sharp Fall 2005 Tuesday, December 20, 2005 4.1 An integrable function f(x) is defined on the interval [0, 1]. Give the formula for the Riemann sum of f(x) on [0, 1] with respect to the partition of [0, 1] into n subintervals of equal length, evaluating the function f(x) at the right endpoint of each subinterval. 4.2 Let r be a positive real number. Use your formula for the Riemann sum to reduce the following limit, n1 � lim kr , n→∞ nr+1 k=1 to a Riemann integral. Then evaluate that Riemann integral (using whatever integration technique you like), and determine the limit. 4.3 Use your formula for the Riemann sum to reduce the following limit, n� 1 lim ,√4k2 + nn→∞ k=1 2 to a Riemann integral. Use an inverse hyperbolic substitution to evaluate the Riemann integral, and determine the limit. You are free to use the following formulas, cosh2(t) − sinh2(t) = 1, d d dt sinh(t) = cosh(t), dt cosh(t) = sinh(t). See also Problem 6.1. 4.4 Compute the following integral, � 2 � 1 t2 e dt − 2 e 4u2 du. 0 0 Hint: It is not possible to write down the antiderivatives of either of the se perate integrands. 4.5 Compute the following integral, π/3 1 dθ. sec(θ) + tan(θ)0 Goal 5. Applications of integration. 5.1 Denote by a the unique angle in the range 0 < a < π/2 satisfying, x − 2 tan(x) =8 − π. 4 Compute the area bounded by the curve, y = tan(x), −π/2 < x < π/2, 318.01 Calculus Jason Starr Final Exam at 9:00am sharp Fall 2005 Tuesday, December 20, 2005 and the tangent line to the curve at (x, y) = (−π/4, −1). 5.2 Find the unique, positive value of h making the area of the region bounded by the parabola, 2 y = x , and the parabola, y = h − x 2 , equal one half the area of the region bounded by the parabola, 2 y = x , and the parabola, y = 1 − x 2 . 5.3 Let a and b be positive real numbers. Find the volume of the solid obtained by rotating about the y-axis the region in the first quadrant of the xy-plane bounded by the x-axis, the y-axis, the line x = aπ and the curve, y = b sin(x/a). You may use either the washer method or the shell method. 5.4 Compute the arc length of the segment of the curve, y = ln(x), bounded by (x, y) = (1, 0) and (x, y) = (e, 1). At some point, it will help to make an inverse substitution. You may make either an inverse trigonometric substitution or an inverse hyperbolic substitution (the resulting integrals are comparable). See also Problem 6.2. 5.5 Compute the area of the surface obtained by rotating about the y-axis the segment of the curve, y = ln(x), bounded by (x, y) = (1, 0) and (x, y) = (e, 1). At some point, it will help to make an inverse substitution. It is best to make an inverse hyperbolic substitution. You are free to use the following formulas, cosh2(t) = 1(cosh(2t) + 1),2 sinh2(t) = 1(cosh(2t) − 1). 2 …


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MIT 18 01 - Final Examination

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