Consider the differentiable function q(x)= (x^ 2 +1 ) x
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the differentiable function
q(x)=ln(x2+1)x,x>0.
Unless otherwise directed, round all numerical answers to three decimal places.
(a) Obtain a high-precision numerical approximation for q′(4).
(b) Without using any symbolic differentiation, approximate q′(4) again by evaluating a symmetric difference quotient with step size h=0.01. Show all calculations.
(c) Compare the two approximations obtained in parts (a) and (b). Determine whether the value from part (a) is accurate to within 0.001 of the true derivative and provide a quantitative justification based on your comparison.