2.6.E: Exercises
- Page ID
- 219870
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The graph of \(y = f(x)\) is shown.
(a) At which integers is \(f\) continuous?
(b) At which integers is \(f\) differentiable?
The graph of \(y = g(x)\) is shown.
(a) At which integers is \(g\) continuous?
(b) At which integers is \(g\) differentiable?
Problems 3 and 4 refer to the values given in this table:
| \(x\) | \(f(x)\) | \(g(x)\) | \(f'(x)\) | \(g'(x)\) | \((f \circ g)(x)\) | \((f \circ g)' (x)\) |
| -2 | 2 | -1 | 1 | 1 | ||
| -1 | 1 | 2 | 0 | 2 | ||
| 0 | -2 | 1 | 2 | -1 | ||
| 1 | 0 | -2 | -1 | 2 | ||
| 2 | 1 | 0 | 1 | -1 |
3. Use the table of values to determine \(( f \circ g )(x)\) and \(( f \circ g )' (x)\) at \(x =\) 1 and 2.
4. Use the table of values to determine \(( f \circ g )(x)\) and \(( f \circ g )' (x)\) at \(x =\) –2, –1 and 0.
5. Use the graphs to estimate the values of \(g(x)\), \(\bf g '(x)\), \((f \circ g)(x)\), \(\mathbf{f '(} g(x) \mathbf{)}\), and \(\mathbf{( f \circ g ) '(} x \mathbf{)}\) at \(x = 1\).
6. Use the graphs to estimate the values of \(g(x)\), \(\bf g '(x)\), \((f \circ g)(x)\), \(\mathbf{f '(} g(x) \mathbf{)}\), and \(\mathbf{( f \circ g ) '(} x \mathbf{)}\) for \(x = 2\).
In problems 7 – 12, find the derivative of each function.
| 7. \(f(x) = (2x – 8)^5\) | 8. \(f(x) = (6x – x^2)^{10}\) | 9. \(f(x) = x \cdot (3x + 7)^5\) |
| 10. \(f(x) = (2x + 3)^6 \cdot (x – 2)^4\) | 11. \(f(x) = \sqrt{x^2 + 6x - 1}\) | 12. \(f(x) = \frac{x-5}{(x+3)^4}\) |
If \(f\) is a differentiable function,
(a) how are the graphs of \(y = f(x)\) and \(y = f(x) + k\) related?
(b) how are the derivatives of \(f(x)\) and \(f(x) + k\) related?