DEPE function

Calculation of the First-Order Derivative of the Explicit Preston Equation

DEPE is used to calculate the first-order derivative of the explicit Preston equation at a given x-value. UTF-8

DEPE(P, x, simpver = NULL)

Arguments

• P: the parameters of the explicit Preston equation or one of its simplified versions.
• x: the x-value used in the explicit Preston equation.
• simpver: an optional argument to use the simplified version of the explicit Preston equation.

Details

When simpver = NULL, the first-order derivative of the explicit Preston equation at a given x-value is selected:

$$f(x)=\frac{b\left[a^{4}\,c_{1}+a^{3}\left(2\,c_{2}-1\right)x+a^2\left(3\,c_{3}-2\,c_{1}\right)x^{2}-3\,a\,c_{2}x^3-4\,c_{3}\,x^{4}\right]}{a^4\sqrt{a^2-x^2}},$$

where P has five parameters: $a$, $b$, $c_{1}$, $c_{2}$, and $c_{3}$.

$\quad$ When simpver = 1, the first-order derivative of the simplified version 1 is selected:

$$f(x)=\frac{b\left[a^{4}\,c_{1}+a^{3}\left(2\,c_{2}-1\right)x-2\,a^2\,c_{1}\,x^{2}-3\,a\,c_{2}x^3\right]}{a^4\sqrt{a^2-x^2}},$$

where P has four parameters: $a$, $b$, $c_{1}$, and $c_{2}$.

$\quad$ When simpver = 2, the first-order derivative of the simplified version 2 is selected:

$$f(x)=\frac{b\left[a^{4}\,c_{1}-a^{3}\,x-2\,a^2\,c_{1}\,x^{2}\right]}{a^4\sqrt{a^2-x^2}},$$

where P has three parameters: $a$, $b$, and $c_{1}$.

$\quad$ When simpver = 3, the first-order derivative of the simplified version 3 is selected:

$$f(x)=\frac{b\left[a^{3}\left(2\,c_{2}-1\right)x-3\,a\,c_{2}x^3\right]}{a^4\sqrt{a^2-x^2}},$$

where P has three parameters: $a$, $b$, and $c_{2}$.

Note

The argument P in the DEPE function has the same parameters, as those in the EPE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .

References

Shi, P., Chen, L., Quinn, B.K., Yu, K., Miao, Q., Guo, X., Lian, M., Gielis, J., Niklas, K.J. (2023) A simple way to calculate the volume and surface area of avian eggs. Annals of the New York Academy of Sciences 1524, 118$-$131. tools:::Rd_expr_doi("10.1111/nyas.15000")

See Also

EPE, fitEPE, SurfaceAreaEPE

Examples

Par3 <- c(4.27, 2.90, 0.0868, 0.0224, -0.0287)
  xx1  <- seq(-4.27, 4.27, by=0.001)
  f1   <- DEPE(P=Par3, x=xx1, simpver=NULL)
  f2   <- -DEPE(P=Par3, x=xx1, simpver=NULL)

  dev.new()
  plot(xx1, f1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-5, 5), ylim=c(-35, 35), xlab=expression(italic(x)), 
       ylab=expression(paste(italic(f), "(", italic(x), ")", sep="")))
  lines(xx1, f2, col=2)  

  graphics.off()