Recent questions in Calculus and Analysis

Differential equations

Transform the second-order differential equation $$\displaystyle{\frac{{{d}^{{{2}}}{x}}}{{{d}{t}^{{{2}}}}}}={3}{x}$$ into a system of first-order differential equations.

Differential equations

True or False? Justify your answer with a proof or a counterexample. You can determine the behavior of all first-order differential equations using directional fields or Euler's method.

Exponential models

Transform the given differential equation or system into an equivalent system of first-order differential equations. $$\displaystyle{x}{''}+{2}{x}'+{26}{x}={82}{\cos{{4}}}{t}$$

Transformations of functions

g is related to one of the parent functions described in Section 1.6. Describe the sequence of transformations from f to g. $$\displaystyle{g{{\left({x}\right)}}}=\sqrt{\frac{1}{4}}{x}$$

Probability and combinatorics

Derivatives

Derivatives

Series

Matrices

For each topic, decide how a scatterplot of the data would likely look. Explain your reasoning. Speed of a runner and amount of time to complete a race

Probability and combinatorics

Series

Derivatives

Find the four second partial derivatives of $$f(x,y)=x^{2}\sin y$$

Probability and combinatorics

$$\displaystyle{\left({3}{t}^{{2}}−{4}{s}\right)}^{{2}}$$

Exponential models

Derivatives

Find derivatives of the functions defined as follows. $$\displaystyle{y}={e}^{{-{2}{x}}}$$

Transformations of functions

Derivatives

Find derivatives for the functions. Assume a, b, c, and k are constants. $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}+{\ln{{x}}}}}}$$

Composite functions

Derivatives