Curve Sketching Mastery
This worksheet provides exercises on curve sketching, focusing on identifying key features of functions and their graphs, suitable for Grade 10 Calculus students.
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Curve Sketching Mastery
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Read each question carefully and provide detailed answers. Show all your work for full credit. For sketching, use the provided coordinate planes or draw your own clearly.
1. For the function f(x) = x³ - 3x² + 2, identify the following:
a) First derivative (f'(x)):
b) Critical points (x-values):
c) Intervals where f(x) is increasing:
d) Intervals where f(x) is decreasing:
2. If the second derivative f''(x) > 0 on an interval, then the function f(x) is concave down on that interval.
True
False
3. A point of inflection occurs where the concavity of a function changes.
True
False
4. A local maximum or minimum occurs at a point where the first derivative is or undefined.
5. To determine the concavity of a function, we use the derivative.
6. Which of the following indicates a local minimum?
f'(x) changes from positive to negative
f'(x) changes from negative to positive
f''(x) < 0
f'(x) = 0 and f''(x) > 0
7. Consider the function g(x) = x⁴ - 4x².
a) Find the x and y-intercepts.
b) Find the critical points and determine if they are local maxima, minima, or neither.
c) Find the intervals of concavity and any inflection points.
d) Use the information above to sketch the graph of g(x) on the coordinate plane below.