Master Calculus 1 Practice Problems with Expert Help

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

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

Calculate the derivatives of all orders:
${\displaystyle f^{\prime }\left(x\right),f{}^{″}\left(x\right),f{}^{‴}\left(x\right),f^4\left(x\right),\dots f^n\left(x\right),\dots f\left(x\right)=e^{2x}}$

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data.

$\begin{array}{|cc|}\hline x& g(x)\\ -1& 6\\ 0& 1\\ 1& 0\\ 2& 3\\ 3& 10\\ \hline\end{array}$

Find the derivatives of the functions.
${\displaystyle r\left(t\right)={\log }_3(t+\sqrt{t})}$

Find the derivatives of the functions.
${\displaystyle u\left(x\right)=\frac{3^{x^2}}{x^2+1}}$

Find the derivatives of the functions.
${\displaystyle f\left(x\right)=e^x{\log }_2\left|x\right|}$

Calculate the derivatives of the functions.
${\displaystyle f\left(x\right)={(4x-1)}^{-1}}$

Find the derivatives:
$f(x)=\frac{x^2+2x+5}{x^2+1}$

Transform the single linear differential equation into a system of first-order differential equations.
${\displaystyle x{}^{″}-3x^{\prime }+2x=t^2}$

Calculate the derivatives of all orders:
${\displaystyle f^{\prime }\left(x\right),f{}^{″}\left(x\right),f{}^{‴}\left(x\right),f^4\left(x\right),\dots f^n\left(x\right),\dots f\left(x\right)=x^4+x^3}$

The table shows the annual service revenues R1 in billions of dollars for the cellular telephone industry for the years 2000 through 2006. Year 2000 2001 2002 2003 2004 2005 2006 R1 52.5 65.3 76.5 87.6 102.1 113.51 25.5
(a) Use the regression capabilities of a graphing utility to find an exponential model for the data. Let t represent the year, with t=10 corresponding to 2000. Use the graphing utility to plot the data and graph the model in the same viewing window.
(b) A financial consultant believes that a model for service revenues for the years 2010 through 2015 is ${\displaystyle R2=6+13+13,9^{0.14}t}$. What is the difference in total service revenues between the two models for the years 2010 through 2015?

Evaluate the line integral, where C is the given curve. $\int y3ds,C÷x=t3,y=t,0?t?3$

Find the derivatives of the functions.
${\displaystyle f\left(x\right)=2x^x-x^2{e}^x}$

Evaluate the line integral, where C is the given curve. C $y^{3}ds,C:x=t^3,y=t,0\le t\le 4$

If $f^{(n)}(0)=(n+1)!\text{ for }n=0,1,2,\dots$, find the Maclaurin series forf and its radius of convergence.

The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada A=33.1e0.009t Uganda A=28.2e0.034t Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013, Ugandas

Evaluate the integral $\int \arctan (x)dx$
A)$-\arctan (x)+\ln (x^2+1)+C$
B)$\frac{1}{x^2+1}+C$
C)$x\arctan (x)-\frac{1}{2}\ln (x^2+1)+C$
D) $\ln |\arccos (x)|+C$
E) $x\arctan (x)+C$

Transform the given differential equation or system into an equivalent system of first-order differential equations.
${\displaystyle t^3{x}^{\left(3\right)}-2t^{2}x{}^{″}+3t{x}^{\prime }+5x=\ln t}$

Solve the higher order derivatives:
y+4y+3y=0