The function f is twice differentiable for x>0 with f(1)=15 and f′′(1)=20 . Values of f′ , the derivative of f , are given for selected values of x in the table above.
a. Write an equation for the line tangent to the graph of f at x=1 . Use this line to approximate f(1.4) .
b. Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate ∫11.4f′(x)dx . Use the approximation for ∫11.4f′(x)dx to estimate the value of f(1.4) . Show the computations that lead to your answer.
c. Use Euler's method, starting at x=1 with two steps of equal size, to approximate f(1.4) . Show the computations that lead to your answer.
d. Write the second-degree Taylor polynomial for f about x=1 . Use the Taylor polynomial to approximate f(1.4) .