# Calculus 1: Analyzing functions questions and answers

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Analyzing functions

### Find the critical points of the following functions. $$\displaystyle{f{{\left({x},\ {y}\right)}}}=-{4}{x}^{{{2}}}+{8}{y}^{{{2}}}-{3}$$

Analyzing functions

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

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

Analyzing functions

### 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

### 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

### 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

### 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

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

Analyzing functions