Notes PDF
More Challenging Problems
1. Find an equation of a function whose derivative is negative everywhere.
Answer
1. For example f = – x.
Solution
2. Find the sign of the slope to x2 at x = -10, 0, and 3.
Answer
1. f (-10) < 0, f (0) = 0, f (3) > 0.
Solution
3. Give an example of a function whose slope is zero at x = 5, and nonzero everywhere else.
Answer
1. f (x) = (x-5)2
Solution
4. If f ‘ (x) < 0 everywhere, does it mean that the function goes towards -∞ for large values of x?
Answer
4. No. Can you think of an example?
Solution
5. Sketch a graph of f (x) = x2 – 2x.
Answer
1. Parabola up, lowest point at (1,-1), crossing x-axis at x = 0 and x = 2
Solution
1. We can see easily that this function is zero at x = 0 and x = 2. Its derivatives is f ‘ (x) = 2x – 2. This is zero at x = 1, negative for x < 1 and positive for x > 1. The function is a parabola, with lowest point at x = 1, going up from there in both directions. Note: this can also be done by completing the square, writing f(x) = (x-1)2 – 1. This is a parabola up with lowest point at (1,-1).