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Calculus Lab Activity Review

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Questions and Answers

In Lab activity 1.2.4, what function is used to find the difference quotient?

$f(x) = x$
$f(x) = x^3$
$f(x) = x^4$
$f(x) = x^2$ (correct)

Which limit expression is used in Lab activity 2.3.4?

$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h^2}$
$\lim_{h \to \infty} \frac{f(a+h) - f(a)}{h}$
$\lim_{a \to 0} \frac{f(a+h) - f(a)}{h}$
$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ (correct)

During the differentiation proof in Lab activity 2.3.4, which algebraic identity is used for rationalizing the denominator?

$a^2 + b^2 = (a+b)^2$
$a^2 - 2ab + b^2 = (a-b)^2$
$a^2 + 2ab + b^2 = (a+b)^2$
$a^2 - b^2 = (a-b)(a+b)$ (correct)

In Lab activity 2.3.4, given $f(x) = \sqrt{x}$, which expression represents $f(a+h)$?

<p>$\sqrt{a+h}$ (A)</p> Signup and view all the answers

In the differentiation process shown, what final expression is obtained for $\frac{d}{dx} \sqrt{x}$?

<p>$\frac{1}{2\sqrt{a}}$ (B)</p> Signup and view all the answers

Flashcards

Difference quotient formula

Used to calculate the instantaneous rate of change of a function. The formula is: (f(a+h) - f(a)) / h, as h approaches 0.

Limit expression for derivative

Represents the instantaneous rate of change of a function at a specific point. It is written as the limit of (f(a + h) - f(a)) / h as h approaches 0.

Algebraic identity for rationalizing denominator

a² - b² = (a-b)(a+b) is used to simplify expressions with square roots in denominators by multiplying by the conjugate

f(a+h) for f(x)=√x

Substituting (a+h) into the function to get √(a+h).

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Derivative of √x

The derivative of the square root function, √x, is 1 / (2√x).