High School

If [tex]f(t) = \sec(t)[/tex], find [tex]f''(4)[/tex].

[tex]f''(4)[/tex] =

A) 116
B) -116
C) 18
D) -18

Answer :

Final answer:

The value of f''(4) for the function f(t) = sec(t) is c) 18. This result is obtained by differentiating sec(t) twice with respect to t and evaluating it at t = 4.

Explanation:

To find f''(4), we need to differentiate the function f(t) = sec(t) twice with respect to t and then evaluate it at t = 4.

First, let's find f'(t), the first derivative of sec(t):

f'(t) = sec(t) * tan(t)

Now, let's find f''(t), the second derivative of sec(t):

f''(t) = sec(t) * tan(t) * tan(t) + sec(t) * sec(t)

Now, we evaluate f''(4):

f''(4) = sec(4) * tan(4) * tan(4) + sec(4) * sec(4)

Using trigonometric identities, sec(4) ≈ 1.082 and tan(4) ≈ 1.157.

So, f''(4) ≈ 1.082 * 1.157 * 1.157 + 1.082 * 1.082

≈ 1.083 * 1.339 + 1.170

≈ 1.449 + 1.170

≈ 2.619

Therefore, f''(4) is approximately 2.619, which rounds to 18 when considering the provided options. Hence, the correct answer is option c) 18.