AP Calculus BC 2.3 Estimating Derivatives of a Function at a Point Exam Style Questions - FRQ
Question
(a) Topic-2.3-Estimating Derivatives of a Function at a Point
(b) Topic-4.1-Interpreting the Meaning of the Derivative in Context
(c) Topic-7.5-Approximating Solutions Using Euler’s Method
(d) Topic-6.11-Integrating Using Integration by Parts BC
5. The function f is twice differentiable for all x with f( 0) = 0. Values of f ‘, the derivative of f , are given in the table for selected values of x.
(a) For x ≥ 0, the function h is defined by \(h(x)=\int_{0}^{x}\sqrt{1+(f’+(t))}^{2}dt\). Find the value of \(h'(\pi )\). Show the work that leads to your answer.
(b) What information does \(\int_{0}^{x}\sqrt{1+(f’+(x))}^{2}dt\) provide about the graph of f ?
(c) Use Euler’s method, starting at x = 0 with two steps of equal size, to approximate \(f(2\pi )\) . Show the computations that lead to your answer.
(d) Find \(\int (t+5)cos(\frac{1}{4})dt\) . Show the work that leads to your answer.
▶️Answer/Explanation
5(a) For x ≥ 0, the function A is defined by \(h(x)=\int_{0}^{x}\sqrt{1+(f’+(t))}^{2}dt\). Find the value of\(h'(\pi )\) . Show the work that leads to your answer.
\(h'(x)=\sqrt{1+(f’+(t))}^{2}\)
\(h'(\pi ) =\sqrt{1+(f’+(t))}^{2}=\sqrt{1+6^{2}}=\sqrt{37}\)
5(b) What information does \(\int_{0}^{\pi }\sqrt{1+(f’+(t))}^{2}dx \)provide about the graph of f ?
\(\int_{0}^{\pi }\sqrt{1+(f’+(t))}^{2}dx \) is the arc length of the graph of f on [0, \(\pi)\)].
5(c) Use Euler’s method, starting at x = 0 with two steps of equal size, to approximate \(f(2\pi )\). Show the computations that lead to your answer.
\(f(\pi )\approx f(0)+\pi f'(0)=0+5\pi =5\pi\)
\(f(2\pi ) \approx f(\pi )+\pi f'(\pi )\)
\(\approx 5\pi +6\pi =11\pi\)
5(d) Find\(\int (t+5)cos(\frac{1}{4})dt\) . Show the work that leads to your answer.