Question

# In each of these cases, find the rate of change of f(t) with respect to t at the given value of t. a. $f(t)=t^{3}-4 t^{2}+5 t \sqrt{t}-5$ at t = 4 b. $f(t)=\frac{2 t^{2}-5}{1-3 t}$ at t = -1

Solution

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$\textbf{\underline{Part A}}$

We can use the basic differentiation rules to find the derivative of $f(t)$.

\begin{align*} f'(t)&=\dfrac{d}{dt}\qty[t^3-4t^2+5t\sqrt{t}-5]\\ &=\dfrac{d}{dt}[t^3]-\dfrac{d}{dt}[4t^2]+\dfrac{d}{dt}[5t^{3/2}]-\dfrac{d}{dt}[5] &{\color{#c34632}\text{Sum/Difference Rule}}\\ &=\dfrac{d}{dt}[t^3]-4\cdot \dfrac{d}{dt}t^2+5\cdot \dfrac{d}{dt}[t^{3/2}-\dfrac{d}{dt}[5] &{\color{#c34632}\text{Constant Multiple Rule}}\\ &=3\cdot t^{(3-1)}-4\cdot 2\cdot t^{(2-1)}+5\cdot \dfrac{3}{2}\cdot t^{(3/2-1/2)}-0 &{\color{#c34632}\text{Power \& Constant Rules}}\\ &=3t^2-8t+\dfrac{15}{2}t^{1/2}\\ &={\color{#c34632}3t^2-8t+\dfrac{15}{2}\sqrt{t}}\\ \end{align*}

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