1. Limits and Continuity

Given the function $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ between $x=1$ and $x=2$?

Determine the interval of convergence for the series $\sum n_1\infty \frac{{(x-2)}^n}{n}$. Express your answer using interval notation.

Suppose a contour map is given for a function $f$, and you are asked to estimate the partial derivatives $f_x(2,1)$ and $f_y(2,1)$ at the point $(2,1)$. Which of the following best describes how you would use the contour map to estimate these partial derivatives?

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos \left(4x\right)}{1+4x-e^{4x}}$.

Use a series expansion to evaluate the limit: $\underset{x\to 0}{lim}\frac{(1+x)-1-\frac{1}{2}{x}^2}{x^2}$. What is the value of this limit?

Given a curve $C$ and a contour map of a function $f$ whose gradient is continuous, which of the following statements is true about the direction of the gradient vector of $f$ at a point on $C$?

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

b. How does the graph behave as x → ±∞?

Give reasons for your answers.

y = (3/2)(x / (x − 1))²/³

6) Use the Intermediate Value Theorem to show that the equation $x^2-6x-3=0$ has a solution on the interval $[0,7]$. Which of the following justifies the existence of a solution?

The graph of $y=f\left(x\right)$ is shown above. What is $\underset{x\to 1}{lim}f\left(x\right)$?

Given the function $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ on the interval $(-1,2)$?

Which of the following best describes the remainder estimate for the integral test when determining the convergence of a series $\sum \underset{\infty }{n=1}{a}_n$ with $a_n=f\left(n\right)$, where $f$ is positive, continuous, and decreasing for $x\ge 1$?

If a function $f$ is continuous on $(-\infty ,\infty )$, which of the following statements is always true?

Which of the following functions is $o\left(x^2\right)$ as $x$ approaches $0$?

Which of the following series can be shown to converge by using the ratio test?

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $f\left(x\right)$ at a point $x=a$?