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Find derivatives for the functions. Assume a, b, c, and k are constants.
${\displaystyle y^{\prime }={(17x+24x^{\frac{1}{2}})}^{\prime }}$

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. [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day 1 and assuming \(α=0.1\) and a carrying capacity of 10 million cells, how long does it take to reach “detection” stage at 5 million cells?

Find derivatives for the functions. Assume a, b, c, and k are constants.

${\displaystyle s^{\prime }\left(t\right){=}^1{\left(6t^{-2}\right)}^{\prime }+{\left(3t^3\right)}^{\prime }-{\left(4t^{\frac{1}{2}}\right)}^{\prime }}$

Linear and quadratic approximation 
1. Find the linear approximating polynomial for the following functions centered at the given point a. 
2. Find the quadratic approximating polynomial for the following functions centered at a. 
3. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. 
${\displaystyle f\left(x\right)=x^{\frac{1}{3}},a=8}$ approximiate ${\displaystyle {7.5}^{\frac{1}{3}}}$

Given ${\displaystyle P\left(x\right)=3x^2+4y^2}$ and ${\displaystyle R\left(x\right)=-7x^2+4xy-3y^2}$, find $P(x)-R(x)$.

Do the polynomials ${\displaystyle x^3-2x^2+1,4x^2-x+3}$, and $3x-2$ generate ${\displaystyle P_3\left(R\right)}$?
Justify your answer.