Lecture 8 power & exp rules
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This document covers derivatives of various functions including: 1) Constant functions have a derivative of 0 since their graph is a horizontal line with zero slope. 2) The power rule states that the derivative of x^n is nx^{n-1}. This can be used to take derivatives of polynomials by applying the constant multiple, sum, and difference rules. 3) Exponential functions like e^x have a derivative of the original function. Higher derivatives are obtained by repeatedly taking derivatives, with the nth derivative denoted as f^(n).
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- Using derivatives to find equations of tangent lines and instantaneous rates of change.
Lemh105
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C:\Fakepath\Stew Cal4e 1 6
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Indefinite Integral
Here are the steps to solve this problem:
(a) Let t = time and y = height. Then the differential equation is:
dy/dt = -32 ft/sec^2
Integrate both sides:
∫dy = ∫-32 dt
y = -32t + C
Initial conditions: at t = 0, y = 0
0 = -32(0) + C
C = 0
Therefore, the equation is: y = -32t
When y = 0 (the maximum height), t = 0.625 sec
(b) Put t = 0.625 sec into the equation:
y = -32(0.625) = -20 ft
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Lecture 8 power & exp rules
- 1. Derivatives of Polynomials and Exponential 3.1 Functions
- 2. 2 Derivative of a Constant Function f (x) = c. The graph is the horizontal line y = c, which has slope 0, so we must have f '(x) = 0. Figure 1 The graph of f (x) = c is the line y = c, so f ¢(x) = 0.
- 3. 3 Derivative of a Constant Function
- 4. 4 Exercise – Constant Multiple Rule Find y’ if y = π
- 5. 5 Power Rule
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- 8. 8 Constant Multiple Rule
- 9. Example 4 – Using the Constant Multiple Rule 9
- 10. 10 Sum Rule
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- 12. 12 Derivatives of Polynomials
- 15. 15 Example 8 If f (x) = ex – x, find f ¢. Compare the graphs of f and f ¢. Solution: Using the Difference Rule, we have
- 16. 16 Example 8 – Solution The function f and its derivative f ¢ are graphed in Figure 8. cont’d Figure 8
- 17. 17 Example 8 – Solution Notice that f has a horizontal tangent when x = 0; this corresponds to the fact that f ¢(0) = 0. Notice also that, for x > 0, f ¢(x) is positive and f is increasing. When x < 0, f ¢(x) is negative and f is decreasing. cont’d
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- 22. Higher Derivatives - Acceleration The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of the object. Thus the acceleration function is the derivative of the velocity function and is therefore the second derivative of the position function: 22 a(t) = v ′(t) = s ′′(t) or, in Leibniz notation,
- 23. 23 Higher Derivatives The process can be continued. The fourth derivative f ′′′′ is usually denoted by f (4). In general, the nth derivative of f is denoted by f (n) and is obtained from f by differentiating n times. If y = f (x), we write
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