# Calculus 1: Analyzing functions questions and answers Recent questions in Analyzing functions
Analyzing functions
ANSWERED ### Find the critical points of the following functions. $$\displaystyle{f{{\left({x},\ {y}\right)}}}=-{4}{x}^{{{2}}}+{8}{y}^{{{2}}}-{3}$$

Analyzing functions
ANSWERED ### What do the $$\lim_{h\to0}\frac{\sin h}{h}$$ and $$\lim_{h\to0}\frac{\cosh-1}{h}$$ have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?

Analyzing functions
ANSWERED ### Which of the following is a verbal version of the Product Law (assuming the limits exist)? (a) The product of two functions has a limit. (b) The limit of the products the product of the limits. (c) The product of a limitis a product of functions. (d) A limit produces a product of functions.

Analyzing functions
ANSWERED ### Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions Use integration by parts to integrate functions. Integrate products of powers of trigonometric functions. Evaluate $$\displaystyle{I}=\int{x}{\sin{{3}}}{x}{\left.{d}{x}\right.}$$

Analyzing functions
ANSWERED ### Find the critical points of the following functions. $$\displaystyle{f{{\left({x},\ {y}\right)}}}={x}^{{{4}}}+{y}^{{{4}}}-{4}{x}-{32}{y}+{10}$$

Analyzing functions
ANSWERED ### In this problem, use the following functions: $$\displaystyle{a}{\left\lbrace{\left({x}\right\rbrace}\right)}=\sqrt{{{x}}}$$ $$\displaystyle{h}{\left\lbrace{\left({x}\right\rbrace}\right)}={x}^{{{2}}}$$ $$\displaystyle{n}{\left\lbrace{\left({x}\right\rbrace}\right)}={e}^{{{x}}}$$ $$\displaystyle{r}{\left\lbrace{\left({x}\right\rbrace}\right)}={x}+{7}$$ $$\displaystyle{p}{\left\lbrace{\left({x}\right\rbrace}\right)}=\frac{{1}}{{x}}$$ $$\displaystyle{g}{\left\lbrace{\left({x}\right\rbrace}\right)}={5}{x}$$ Each function below is some composition of the functions above. For example, the function $$\displaystyle{y}={\left\lbrace{\left({x}+{7}\right\rbrace}\right)}^{{{2}}}$$ is the composition $$\displaystyle{h}{\left\lbrace{\left({r}{\left\lbrace{\left({x}\right\rbrace}\right)}\right\rbrace}\right)}$$ For each function below, write the simplest composition using two of the functions from the given list. $$\displaystyle{5}{\left\lbrace{\left(\frac{{1}}{{x}}\right\rbrace}\right)}=$$ $$\displaystyle{5}{\left({e}^{{{x}}}\right)}=$$ $$\displaystyle\sqrt{{{\left\lbrace{\left({x}+{7}\right\rbrace}\right)}}}=$$ The next two functions are compositions of three different functions from the given list. $$\displaystyle{\left\lbrace{\left({\left\lbrace{\left(\frac{{1}}{{x}}\right\rbrace}\right)}+{7}\right\rbrace}\right)}^{{{2}}}=$$ $$\displaystyle{e}^{{{\left\lbrace{\left({x}^{{2}}\right\rbrace}\right)}}}+{7}=$$

Analyzing functions
ANSWERED ### Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. $$\displaystyle{f{{\left({x},\ {y}\right)}}}={\sin{}}$$$$(2\pi x)\cos(\pi y)$$, for $$\displaystyle{\left|{x}\right|}\leq{\frac{{{1}}}{{{2}}}}$$ and $$\displaystyle{\left|{y}\right|}\leq{\frac{{{1}}}{{{2}}}}$$

Analyzing functions
ANSWERED ### Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. $$\displaystyle{f{{\left({x},\ {y}\right)}}}={\left({4}{x}-{1}\right)}^{{{2}}}+{\left({2}{y}+{4}\right)}^{{{2}}}+{1}$$

Analyzing functions
ANSWERED ### Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}$$

Analyzing functions
ANSWERED ### To determine (1) If f and g are both even functions, whether $$\displaystyle{f}+{g}$$ is even? (2) If f and g are both odd functions, whether $$\displaystyle{f}+{g}$$ is odd? (3) If f is even and g is odd function, what will $$\displaystyle{f}+{g}$$?

Analyzing functions
ANSWERED ### What do the limits $$\lim_{h\to0}\frac{\sin h}{h}$$ and $$\lim_{h\to0}\frac{\cos h-1}{h}$$ have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?

Analyzing functions
ANSWERED ### What are the derivatives of the six basic hyperbolic functions? What are the corresponding integral formulas? What similarities do you see here to the six basic trigonometric functions?

Analyzing functions
ANSWERED ### Inverse Trigonometric Functions Are the derivatives of the inverse trigonometric functions algebraic or transcendental functions? Explain.

Analyzing functions
ANSWERED ### Analyzing critical points. Use the Second Derivative Test to classify the critical points of $$\displaystyle{f{{\left({x},\ {y}\right)}}}={x}^{{{2}}}+{2}{y}^{{{2}}}-{4}{x}+{4}{y}+{6}$$.

Analyzing functions
ANSWERED ### Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions Use integration by parts to integrate functions. Integrate products of powers of trigonometric functions. Evaluate $$\displaystyle\int{{\sin}^{{{4}}}\theta}{{\cos}^{{{2}}}\theta}{d}\theta$$

Analyzing functions
ANSWERED ### Analizing the polynomial $$\displaystyle{y}={\left({x}-{1}\right)}{\left({3}{x}-{2}\right)}{\left({x}+{3}\right)}$$ a) Determine the roots of the polinomial b) Determine the end behaviors c) Determine how many turning points

Analyzing functions
ANSWERED ### Find functions fand g such that $$\displaystyle{h}={g}\circ{f}$$ (Note: The answer is not unique. Enter your answers as a comma-separated list of functions. Use non- identity functions for fand g.) $$h\left(x\right)=\left(5x^{2}-8\right)^{-3}$$

Analyzing functions
ANSWERED ### Without integrating, explain why $$\displaystyle{\int_{{-{2}}}^{{{2}}}}{x}{\left({x}^{{{2}}}+{1}\right)}^{{{2}}}{\left.{d}{x}\right.}={0}$$

Analyzing functions
ANSWERED ### Find the critical points of the following functions. $$\displaystyle{f{{\left({x},\ {y}\right)}}}={y}{e}^{{{x}}}-{e}^{{{y}}}$$
ANSWERED 