Let $F ( x ) = \int _ { 5 } ^ { x } 5 t ^ { 2 } d t$

question 54

Short Answer

Let $F ( x ) = \int _ { 5 } ^ { x } 5 t ^ { 2 } d t$ .
a. Use Part 1 of the Fundamental Theorem of Calculus to find $F ^ { \prime } ( x )$ .
b. Use Part 2 of the Fundamental Theorem of Calculus to integrate $\int _ { 5 } ^ { x } 5 t ^ { 2 } d t$ to obtain an alternative expression for $F ( x )$ .
c. Differentiate the expression for $F ( x )$ found in part (b). The Fundamental Theorem of Calculus, Part 1 If $f$ is continuous on $[ a , b ]$ , then the function $F$ defined by $F ( x ) = \int _ { a } ^ { x } f ( t ) d t \quad a \leq x \leq b$ is differentiable on $( a , b )$ , and $F ^ { \prime } ( x ) = \frac { d } { d x } \int _ { a } ^ { x } f ( t ) d t = f ( x )$ The Fundamental Theorem of Calculus, Part 2
If $f$ is continuous on $[ a , b ]$ , then $\int _ { a } ^ { b } f ( x ) d x = F ( b ) - F ( a )$ where $F$ is any antiderivative of $f$ , that is, $F ^ { t } = f$ .

---

Definitions:

Auditory Cortex

The part of the brain that is responsible for processing auditory information, situated in the temporal lobe and critical for interpreting sounds and language.

Temporal Lobes

Sections of the brain located on either side of the head, above the ears, involved in processing auditory information and encoding memory.

Primary Visual Cortex

The part of the brain in which visual information is processed.

Visual Information

Data perceived through the sense of sight, which includes pictures, symbols, charts, and all visual media.