Question

use part i of the fundamental theorem of calculus to find each of the following when $f(x)=\int_{5}^{x} t^{5}dt$.

$f(x)=\square$
$f(-2)=\square$

Explanation:

Step1: Apply Fundamental Theorem of Calculus Part I

The Fundamental Theorem of Calculus Part I states that if \( f(x)=\int_{a}^{x}g(t)dt \), then \( f^{\prime}(x) = g(x) \). Here, \( g(t)=t^{5} \), so by the theorem, \( f^{\prime}(x)=x^{5} \).

Step2: Evaluate \( f^{\prime}(-2) \)

Substitute \( x = - 2 \) into \( f^{\prime}(x)=x^{5} \). We know that \( (-2)^{5}=(-2)\times(-2)\times(-2)\times(-2)\times(-2)=-32 \).

Answer:

\( f^{\prime}(x)=\boldsymbol{x^{5}} \)
\( f^{\prime}(-2)=\boldsymbol{-32} \)