Grade 12 Math Performance Tasks
This worksheet contains advanced math performance tasks for Grade 12 students, covering functions, calculus concepts, and problem-solving.
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Grade 12 Math Performance Tasks
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Read each performance task carefully and provide complete solutions, showing all your work and reasoning. Use appropriate mathematical notation and terminology.
1. A company manufactures cylindrical cans to hold a volume of 500 cm³. The cost of the material for the top and bottom of the can is $0.02 per cm², and the cost of the material for the side of the can is $0.01 per cm². Determine the dimensions (radius and height) of the can that minimize the total cost of the materials. Show all your steps, including the objective function, constraint, and the use of calculus to find the minimum.
2. A ladder 10 meters long is leaning against a vertical wall. The bottom of the ladder is pulled away from the wall at a rate of 0.5 m/s. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 meters from the wall? Draw a diagram and clearly label your variables.
3. Consider the function f(x) = x³ - 6x² + 9x - 1. Which of the following statements about the function is TRUE?
The function has a local maximum at x = 3.
The function is decreasing on the interval (1, 3).
The function has an inflection point at x = 1.
The function is concave up on the interval (-∞, 2).
4. Find the area of the region bounded by the curves y = x² and y = 2x - x². Include a sketch of the region and show the integral setup.
5. For the function g(x) = (x² - 4) / (x - 2), the limit as x approaches 2 is . This function has a discontinuity at x = 2.