2.7.E: Exercises
- Page ID
- 157569
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In problems 1 and 2, each quotation is a statement about a quantity of something changing over time. Let \(f(t)\) represent the quantity at time \(t\). For each quotation, tell what \(f\) represents and whether the first and second derivatives of \(f\) are positive or negative.
1. (a) "Unemployment rose again, but the rate of increase is smaller than last month."
(b) "Our profits declined again, but at a slower rate than last month."
(c) "The population is still rising and at a faster rate than last year."
2. (a) "The child's temperature is still rising, but slower than it was a few hours ago."
(b) "The number of whales is decreasing, but at a slower rate than last year."
(c) "The number of people with the flu is rising and at a faster rate than last month."
On which intervals is the function in the graph (a) concave up? (b) concave down?
On which intervals is the function in graph (a) concave up? (b) concave down?
Sketch the graphs of functions which are defined and concave up everywhere and which have
(a) no roots.
(b) exactly 1 root.
(c) exactly 2 roots.
(d) exactly 3 roots.
In problems 6 – 9, a function and values of \(x\) so that \(f '(x) = 0\) are given. Use the Second Derivative Test to determine whether each point \((x, f(x))\) is a local maximum, a local minimum or neither
| 6. \(f(x) = 2x^3 – 15x^2 + 6, x = 0, 5 \). |
| 7. \(g(x) = x^3 – 3x^2 – 9x + 7, x = –1, 3 \). |
| 8. \(h(x) = x^4 – 8x^2 – 2, x = –2, 0, 2 \). |
| 9. \(f(x) = x \cdot \ln(x), x = 1/e \). |
Which of the labeled points in the graph are inflection points?
Which of the labeled points in the graph are inflection points?
How many inflection points can a
(a) quadratic polynomial have?
(b) cubic polynomial have?
(c) polynomial of degree \(n\) have?
Fill in the table with "+", "–", or "0" for the function shown.
| \(x\) | \(f(x)\) | \(f'(x)\) | \(f''(x)\) |
| 0 | |||
| 1 | |||
| 2 | |||
| 3 |
Fill in the table with "+", "–", or "0" for the function shown.
| \(x\) | \(g(x)\) | \(g'(x)\) | \(g''(x)\) |
| 0 | |||
| 1 | |||
| 2 | |||
| 3 |
In problems 15 – 21, find the derivative and second derivative of each function.
| 15. \(f(x) = 7x^2 + 5x – 3\) |
| 16. \(f(x) = (2x – 8)^5\) |
| 17. \(f(x) = (6x – x^2)^{10}\) |
| 18. \(f(x) = x \cdot (3x + 7)^5 \) |
| 19. \(f(x) = (2x^3 + 3)^6\) |
| 20. \(f(x) = \sqrt{x^2 + 6x - 1}\) |
| 21. \(f(x) = \ln (x^2+4)\) |