exp_2: CppAD Forward and Reverse Sweeps
.

Purpose
Use CppAD forward and reverse modes to compute the partial derivative with respect to $x$, at the point $x = .5$, of the function
exp_2(
x)
as defined by the exp_2.hpp include file.

Exercises
1. Create and test a modified version of the routine below that computes the same order derivatives with respect to $x$, at the point $x = .1$ of the function
exp_2(
x)
2. Create a routine called
exp_3(
x)
that evaluates the function $$f(x) = 1 + x^2 / 2 + x^3 / 6$$ Test a modified version of the routine below that computes the derivative of $f(x)$ at the point $x = .5$.

# include "exp_2.hpp"        // second order exponential approximation
{     bool ok = true;
using CppAD::vector;    // can use any simple vector template class
using CppAD::NearEqual; // checks if values are nearly equal

// domain space vector
size_t n = 1; // dimension of the domain space
X[0] = .5;    // value of x for this operation sequence

// declare independent variables and start recording operation sequence

// evaluate our exponential approximation

// range space vector
size_t m = 1;  // dimension of the range space
Y[0] = apx;    // variable that represents only range space component

// Create f: X -> Y corresponding to this operation sequence
// and stop recording. This also executes a zero order forward
// sweep using values in X for x.

// first order forward sweep that computes
// partial of exp_2(x) with respect to x
vector<double> dx(n);  // differential in domain space
vector<double> dy(m);  // differential in range space
dx[0] = 1.;            // direction for partial derivative
dy    = f.Forward(1, dx);
double check = 1.5;
ok   &= NearEqual(dy[0], check, 1e-10, 1e-10);

// first order reverse sweep that computes the derivative
vector<double>  w(m);   // weights for components of the range
vector<double> dw(n);   // derivative of the weighted function
w[0] = 1.;              // there is only one weight
dw   = f.Reverse(1, w); // derivative of w[0] * exp_2(x)
check = 1.5;            // partial of exp_2(x) with respect to x
ok   &= NearEqual(dw[0], check, 1e-10, 1e-10);

// second order forward sweep that computes
// second partial of exp_2(x) with respect to x
vector<double> x2(n);     // second order Taylor coefficients
vector<double> y2(m);
x2[0] = 0.;               // evaluate second partial .w.r.t. x
y2    = f.Forward(2, x2);
check = 0.5 * 1.;         // Taylor coef is 1/2 second derivative
ok   &= NearEqual(y2[0], check, 1e-10, 1e-10);

// second order reverse sweep that computes
// derivative of partial of exp_2(x) w.r.t. x
dw.resize(2 * n);         // space for first and second derivatives
dw    = f.Reverse(2, w);
check = 1.;               // result should be second derivative
ok   &= NearEqual(dw[0*2+1], check, 1e-10, 1e-10);

return ok;
}