MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
x
1
1.1
1.2
1.3
1.4
f′(x)
8
10
12
13
14.5
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.
(1) Write an equation for the line tangent to the graph of f at x=1 . Use this line to approximate f(1.4) .
(2) 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.
(3) 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.
(4) Write the second-degree Taylor polynomial for f about x=1 . Use the Taylor polynomial to approximate f(1.4) .