Consider the functionand let be a positive real number. For , let be a point on the curve . Let denote the value of such that the distance between the foot of the perpendicular from to the -axis and the point where the tangent to the curve at meets the -axis is equal to . Evaluate[4 points]\subsection*{Numerical answer}If the value is , compute .\
[THOUGHT] I'll define f, compute the tangent intersection distance expression t(s)=s f'(s), invert relationship via change of variables, and perform the integral symbolically with Python, then express result as p/q + ln 2 and compute p+q. [/THOUGHT] [PYTHON] import sympy as sp
s = sp.symbols('s', positive=True) x = sp.symbols('x', positive=True)
Define f and its derivative
f = sp.Rational(1,2)*x**2 - x + sp.log(1 + x) fp = sp.diff(f, x)
Check derivative sign for x>0 quickly (sample)
sample_vals = [0.1, 0.5, 1, 2, 5, 10] print("f'(x) at samples:") for val in sample_vals: print(val, float(fp.subs(x, val)))
Define t(s) = s f'(s)
t = s * fp.subs(x, s) t_simplified = sp.simplify(t) print("t(s) =", t_simplified)
Compute t'(s)
tprime = sp.diff(t_simplified, s) tprime_simplified = sp.simplify(tprime) print("t'(s) =", tprime_simplified)
Solve t(s) = 1/2 and t(s) = 27/4 to find s1, s2
eq1 = sp.Eq(t_simplified, sp.Rational(1,2)) eq2 = sp.Eq(t_simplified, sp.Rational(27,4)) s1_solutions = sp.solve(eq1, s) s2_solutions = sp.solve(eq2, s) print("Solutions for t=1/2:", s1_solutions) print("Solutions for t=27/4:", s2_solutions)
Filter positive solutions
s1_pos = [sol for sol in s1_solutions if sol.is_real and sol>0] s2_pos = [sol for sol in s2_solutions if sol.is_real and sol>0] print("Positive s1:", s1_pos) print("Positive s2:", s2_pos)
s1 = sp.simplify(s1_pos[0]) s2 = sp.simplify(s2_pos[0]) print("Chosen s1, s2:", s1, s2)
Now compute integral I = ∫_{1/2}^{27/4} g(t) dt
With t = t(s), g(t) = s, dt = t'(s) ds
So I = ∫_{s1}^{s2} s * t'(s) ds
I = sp.integrate(s * tprime_simplified, (s, s1, s2)) I_simplified = sp.simplify(I) print("Integral I =", I_simplified)
Express I in form p/q + ln(2)
p, q = sp.symbols('p q', rational=True)
Separate rational and log components
I_rational_part = sp.simplify(sp.nsimplify(I_simplified - sp.log(2))) print("I - ln(2) =", I_rational_part)
Now write I_rational_part as p/q exactly
I_rational_part_q = sp.nsimplify(I_rational_part) print("Rational part:", I_rational_part_q)
Extract numerator and denominator
p_val = sp.Rational(I_rational_part_q).p q_val = sp.Rational(I_rational_part_q).q print("p =", p_val, "q =", q_val, "p+q =", p_val + q_val) [/PYTHON]
[THOUGHT] Using the confirmed Python result, the integral equals log(2) + 157/12, so p=157 and q=12, giving p+q=169. \boxed{169}