⧼exchistory⧽
13 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
Find the derivative with respect to [math]x[/math] of each of the following functions.
- [math]4^{x+1}[/math]
- [math]\log_{10}(x^2+1)[/math]
- [math]\log_{10}4^{x+1}[/math]
- [math]e^{x^2+x+2}[/math]
- [math]xa^x[/math]
- [math]2^xx^2[/math]
- [math]xe^{-x}[/math]
- [math]log_4 (x^2-4^x)[/math]
- [math]x^{x-1}[/math]
- [math]x^{(x^2)}[/math]
- [math](x^x)^2[/math].
BBy Bot
Nov 03'24
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[/math]
If [math]a[/math] and [math]b[/math] are positive numbers not equal to [math]1[/math], prove that [math]\log_ab = \frac1{\log_ba}[/math].
BBy Bot
Nov 03'24
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[/math]
Prove that
- [math]\ln x = (\ln a)(log_a x)[/math]
- [math]\ln x = \frac{\log_ax}{\log_ae}[/math].
BBy Bot
Nov 03'24
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[/math]
Integrate each of the following.
- [math]\int 7^x \; dx[/math]
- [math]\int x^22^{3x^3+4} \; dx[/math]
- [math]\int \frac1{x+2} \ln |x+2| \; dx[/math]
- [math]\int \frac1x \ln \left|\frac1x\right| \; dx[/math]
- [math]\int \log_2e^{7x-5} \; dx[/math]
- [math]\int \frac1{x+3} 3^{\ln |x+3|} \; dx[/math]
- [math]\int e^x5^{(e^x)} \; dx[/math].
BBy Bot
Nov 03'24
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[/math]
- lab{5.4.5a} Differentiate logarithmically [math]y = \sqrt{\frac{(x-1)(x+3)}{(x+2)(x-4)}}[/math].
- For what values of [math]x[/math] is the differentiation in \ref{ex5.4.5a} valid?
BBy Bot
Nov 03'24
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[/math]
If [math]u[/math] is a positive function of [math]x[/math] and [math]a[/math] is positive but not equal to [math]1[/math], show that [math]\log_au = \frac{\ln u}{\ln a}[/math].
BBy Bot
Nov 03'24
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[/math]
Differentiate each of the following functions with respect to [math]x[/math].
- [math]x^{\ln x}[/math]
- [math](\ln x)^x[/math]
- [math](e^x)^{x^2+1}[/math]
- [math](\ln x)^{\ln x}[/math].
BBy Bot
Nov 03'24
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[/math]
- Given only that [math]\log_a1=0[/math] and that [math]\log_apq = \log_ap +\log_aq[/math], prove that [math]\log_a \left(\frac1p \right) = -\log_ap[/math].
- Then prove that [math]\log_a \frac{p}{q} = \log_ap - \log_aq[/math].
BBy Bot
Nov 03'24
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[/math]
Prove all the properties listed in \ref{thm 5.4.7} (see Problem Exercise).
BBy Bot
Nov 03'24
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[/math]
Using the Mean Value Theorem, Theorem, prove that if [math]f^\prime(x) \gt 0[/math] for all [math]x[/math], then [math]f[/math] is a strictly increasing function.