Master Calculus 1 Practice Problems with Expert Help

The following question consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. When does population increase the fastest in the threshold logistic equation ${\displaystyle P^{\prime }\left(t\right)=rP(1-\frac{P}{K})(1-\frac{T}{P})?}$

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)
${\displaystyle 7{\sin }^8\left(x\right)\cos \left(x\right)\ln \left(\sin \left(x\right)\right)dx}$
no. 101. ${\displaystyle \int u^n\ln udu=\frac{u^{n+1}\left\{{(n+1)}^2\right\}[(n+1)\ln u-1]+C}{}}$

Compute the derivatives.
${\displaystyle \frac{d}{dx}\left(x\sec hx\right)}$

Compute the derivatives.
${\displaystyle \frac{d^6}{{dx}^6}\left(\cos hx\right)}$

Differentiate the linear and non-linear equation, and give a reason that why its linear and non-linear.

$xy+z=2$

Find the derivatives.
by differentiating the integral directly.
${\displaystyle \frac{d}{d0}{\int }_0^{\tan 0}{\sec }^{2}ydy}$

Differentiate the linear and non-linear equation, and give a reason that why its linear and non-linear.

$2x(y-3z)=1$

Find the first partial derivatives of the following functions.
${\displaystyle g(u,v)=u\cos v-v\sin u}$

Find The Exponential Function ${\displaystyle f\left(x\right)=a^x}$ Whose Graph Is Given.

Let ${\displaystyle f\left(x\right)={(1+x^2)}^4}$. Find f'(x) in two ways, first by expanding to get an expression for f(x) as a polynomial in x and then differentiating, and then by using the power rule. Are the answers the same?

Find the indicative derivative of the following:
${\displaystyle xy=e^{2x}.y^{\prime }}$

Compute the derivatives. Use the fact that ${\displaystyle e^2+x=e^2}$. ex to find ${\displaystyle \frac{d}{dx}\left(e^{x+2}\right)}$

Differentiate the linear and non-linear equation, and give a reason that why its linear and non-linear.

$\frac{1}{2}x+y-z=\sqrt{7}$

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
$\begin{array}{|cccccccccccc|}\hline x& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13\\ f(x)& 13.98& 17.84& 20.01& 22.7& 24.1& 26.15& 27.37& 28.38& 29.97& 31.07& 31.43\\ \hline\end{array}$

Let ${\displaystyle f\left(x\right)=(x^2+1)(x^3+3)}$. Find f'(x) in two ways:
(a) by multiplying and then differentiating,
(b) by using the product rule.

Graph each function and tell whether it represents exponential growth, exponential decay, or neither. ${\displaystyle y={\left(2.5\right)}^x}$

Transform the given differential equation or system into an equivalent system of first-order differential equations.
${\displaystyle x{}^{″}+2x^{\prime }+26x=82\cos 4t}$

Use derivatives to prove that if ${\displaystyle n\ge 1}$, then ${\displaystyle {(1+x)}^n>1+nx}$ for -1

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

A researcher is trying to determine the doubling time fora population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. ​
$\begin{array}{|cc|}\hline \text{Time (hours)}& \text{Bacteria count (CFU/mL)}\\ 0& 37\\ 4& 47\\ 8& 63\\ 12& 78\\ 16& 105\\ 20& 130\\ 24& 173\\ \hline\end{array}$ ​
Use a graphing calculator to find an exponential curve $f(t)=a\times b^t$ that models the bacteria population t hours later.