Skills Review for The Fundamental Theorem of Calculus

Learning Outcomes

Evaluate a polynomial function at a monomial or binomial value

In the Fundamental Theorem of Calculus section, we will learn how to evaluate definite integrals. Here we will review evaluating functions that have variables raised to powers.

Evaluate Functions Containing Variables Raised to Powers

When given the function $f(x)=x^2+7$ , if asked to find the value of $f(-3t)$ , you would take the variable $x$ in the function and replace it with $-3t$ . So,

$f(-3t)=(-3t)^2+7=9t^2+7$

Note: It is important that the $-3$ and the $t$ both be squared.

Example: Evaluating Functions Containing Variables Raised to Powers

Given $f(x)=x^3+2x-3$ , find $f(-2t)$ .

Show Solution

Everywhere we see $x$ in the function, we replace it with $-2t$ .

So, we have:

$f(-2t)=(-2t)^3+2(-2t)-3=-8t^3-4t-3$

Example: Evaluating Functions Containing Variables Raised to Powers

Given $f(x)=2x^3-9$ , find $f(5t^2)$ .

Show Solution

Everywhere we see $x$ in the function, we replace it with $5t^2$ .

So, we have:

$f(5t^2)=2(5t^2)^3-9=2(125t^6)-9=250t^6-9$

Note: In the above example, when we had to find $(t^2)^3$ , raising a power to a power requires that we multiply the exponents together to give us $t^6$ .

Try It

Given $f(x)=x^2-3x+1$ , find $f(-4t^4)$ .

Show Solution

$f(-4t^4)=16t^8+12t^4+1$

Integration: Prerequisite Material