Calculus Functions and Approximations

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Questions and Answers

What is the linear approximation of a function at a given point?

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.

How do you apply L'Hôpital's Rule to evaluate indeterminate limits?

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.

What does the Intermediate Value Theorem (IVT) state about continuous functions?

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 $f(c) = N$.

Describe the relationship between the first and second derivative in determining concavity.

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

What feature of a function can be inferred from changes in the sign of its first derivative?

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. Signup and view all the answers

How can you identify the original function from its derivative graph?

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. Signup and view all the answers

What is the result of applying the Power Rule of Integration to $x^n$?

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

Explain the process of using $u$-substitution in finding antiderivatives.

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. Signup and view all the answers

How can you approximate the area under a curve using a Riemann sum?

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. Signup and view all the answers

Flashcards

Linear Approximation

An approximation of a function's value near a given point using its tangent line.

L'Hôpital's Rule

A method used to evaluate indeterminate limits of the form 0/0 or ∞/∞ by taking derivatives of the numerator and denominator.

Existence Theorems

Theorems that guarantee the existence of specific points on a function's graph based on certain conditions.

Intervals of Increase/Decrease and Concavity

The process of finding where a function is increasing or decreasing, and where its graph curves upwards or downwards.