# Recent questions in Composite functions

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{y}={4}^{{-{5}{x}+{2}}}$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{y}={3}\dot{{\lbrace}}{4}^{{{x}^{{{2}}}+{2}}}$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{f{{\left({z}\right)}}}={\left({2}{z}+{e}^{{-{z}^{{{2}}}}}\right)}^{{{2}}}$$

Composite functions

### Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as $$\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}.{f{{\left({x},{y}\right)}}}={1}-{\frac{{{\cos{{\left({x}^{{{2}}}+{y}^{{{2}}}\right)}}}}}{{{x}^{{{2}}}+{y}^{{{2}}}}}}$$

Composite functions

### Write an integral that requires three applications of integration by parts. Explain why three applications are needed.

Composite functions

### Find derivatives of the functions defined as follows. $$s=5 \cdot 2^{\sqrt{t-2}}$$

Composite functions

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim{\left({x},{y},{z}\right)}→{\left(-{2},{1},{0}\right)}{x}{e}^{{y}}{z}$$

Composite functions

### Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as $$\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}.{f{{\left({x},{y}\right)}}}={e}^{{{x}{y}}}$$

Composite functions

### Use limit laws and continuity properties to evaluate the limit. $$\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({4},-{2}\right)}}}{x}\sqrt{{{3}}}{\left\lbrace{y}^{{{3}}}+{2}{x}\right\rbrace}$$

Composite functions

### Sketch the graph of the function a) $$f(1)=[[1]]+[[-1]]=1\pm1=0$$ $$f(0)=0$$ $$f(\frac{1}{2})=0\pm1=-1$$ $$f(-2.7)=-3+2=-1$$ b) $$\lim_{x \rightarrow 1}f(x)=-1$$ $$\lim_{x \rightarrow 1+}f(x)=-1$$ $$\lim_{x \rightarrow \frac{1}{2}}f(x)=-1$$ c) f is continuous for all real numbers except $$0,\pm 1,\pm 2,\pm 3, \pm 4\cdots$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{p}={\frac{{{1000}}}{{{9}+{4}{e}^{{-{0.2}{t}}}}}}$$

Composite functions

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim{\left({x},{y}\right)}→{\left({0},{0}\right)}{x}+{4}{y}+{1}$$

Composite functions

### Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}+\sqrt{{{7}-{x}}},{a}={4}$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{y}=-{10}^{{{3}{x}^{{{2}}}-{4}}}$$

Composite functions

### Given that $$\displaystyle{f{{\left({x},{y},{z}\right)}}}={x}{y}+{z},$$ $$\displaystyle{x}={s}^{{2}},$$ $$\displaystyle{y}={s}{t},$$ $$\displaystyle{z}={t}^{{2}},$$ find the composite function.

Composite functions

### Given $$f(x) = 5x − 5$$ and $$g(x) = 5x − 1,$$ Evaluate the composite function $$g[f(0)]$$

Composite functions

### The number of electric scooters e that a factory can produce per day is a function of the number of hours h it operates and is given by $$\displaystyle{e}{\left({h}\right)}={290}{h},{0}\le{h}\le{10}.$$ The daily cost c to manufacture e electric scooters is given by the function $$\displaystyle{c}{\left({e}\right)}={0.05}{e}^{{2}}+{65}{e}+{1000}.$$ (a) Find $$\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}.$$ (b) Evaluate $$\displaystyle{\left({c}\circ{e}\right)}{\left({13}\right)}.$$

Composite functions

### Rewrite the given pair of functions as one composite form $$\displaystyle{g{{\left({x}\right)}}}=\sqrt{{{5}{x}^{{2}}}}$$ $$\displaystyle{x}{\left({w}\right)}={2}{e}^{{w}}$$ $$g(x(ww))=?$$ Evalute the composite function at 1. $$g(x(1))=?$$

Composite functions