Calculus Functions and Approximations

The linear approximation of a function at a point is given by the formula $L(x) = f(a) + f'(a)(x - a)$, where $a$ is the point of tangency.

L'Hôpital's Rule states that for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, you can take the derivative of the numerator and the derivative of the denominator, then re-evaluate the limit.

The IVT states that if $f$ is continuous on the interval $[a, b]$ and $N$ is a number between $f(a)$ and $f(b)$, there exists at least one $c$ in $(a, b)$ such that $f(c) = N$.

<p>The first derivative $f'(x)$ indicates intervals of increase/decrease, while the second derivative $f''(x)$ determines concavity; if $f''(x) &gt; 0$, the function is concave up, and if $f''(x) &lt; 0$, it is concave down.</p> Signup and view all the answers

<p>When the first derivative $f'(x)$ changes from positive to negative, the function $f(x)$ has a relative maximum; when it changes from negative to positive, it has a relative minimum.</p> Signup and view all the answers

<p>You can identify the original function by analyzing the characteristics of its derivative graph: where $f'(x)$ is positive, $f(x)$ is increasing, where it is negative, $f(x)$ is decreasing, and where it crosses the x-axis indicates the critical points.</p> Signup and view all the answers

<p>The Power Rule states that the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.</p> Signup and view all the answers

<p>In $u$-substitution, you choose a substitution $u = g(x)$, differentiate to find $du$, replace variables in the integral, and then integrate in terms of $u$ before substituting back to the original variable.</p> Signup and view all the answers

<p>You can approximate the area under a curve using a Riemann sum by dividing the interval into $n$ subintervals, evaluating the function at specific points (left, right, midpoint), and summing the products of these values with the subinterval widths.</p> Signup and view all the answers