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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.
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