If h(x)=4x3-5x+7 is the derivative of f(x), then \(\lim\limits_{t\rightarrow0}\frac{f(1+t)-f(1)}{t}\) is equal to
- 5
- 6
- 7
- 8
- 0
The Correct Option is B
Solution and Explanation
We are given that \( h(x) = 4x^3 - 5x + 7 \) is the derivative of \( f(x) \), and we need to find the value of:
\( \lim\limits_{t \rightarrow 0} \frac{f(1+t) - f(1)}{t} \).
From the definition of the derivative, we know that:
\( \lim\limits_{t \rightarrow 0} \frac{f(1+t) - f(1)}{t} = f'(1) \).
We are given that \( h(x) = f'(x) \), so:
\( f'(1) = h(1) \).
Now, substitute \( x = 1 \) into the expression for \( h(x) \):
\( h(1) = 4(1)^3 - 5(1) + 7 \).
\( h(1) = 4 - 5 + 7 = 6 \).
The correct answer is 6.
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