5. (5pts) find the slope ( y(x) ) of the tangent line to the curve of ( y(x)=a x^{2}+b x+c ) where ( a, b )…

5. (5pts) find the slope ( y(x) ) of the tangent line to the curve of ( y(x)=a x^{2}+b x+c ) where ( a, b ), and ( c ) are constants. recall that the slope of the tangent line is given by ( y(x)=\frac{y(x+delta x)-y(x)}{delta x} ) then let ( delta x \to 0 ) show all of your work. 6. (5pts) similarly, the velocity ( v(t) ) is the instantaneous rate of change (slope of the tangent line) of ( x(t) ) ( v(t)=\frac{x(t+delta t)-x(t)}{delta t} ) then let ( delta t \to 0 ) find the velocity ( v(t) ) given that ( x(t)=3 t^{2}+4 t ).

5. (5pts) find the slope ( y(x) ) of the tangent line to the curve of ( y(x)=a x^{2}+b x+c ) where ( a, b ), and ( c ) are constants. recall that the slope of the tangent line is given by ( y(x)=\frac{y(x+delta x)-y(x)}{delta x} ) then let ( delta x \to 0 ) show all of your work. 6. (5pts) similarly, the velocity ( v(t) ) is the instantaneous rate of change (slope of the tangent line) of ( x(t) ) ( v(t)=\frac{x(t+delta t)-x(t)}{delta t} ) then let ( delta t \to 0 ) find the velocity ( v(t) ) given that ( x(t)=3 t^{2}+4 t ).

Answer

Explanation:

Step1: Substitute (x(t)=3t^{2}+4t) into the velocity formula

$$ \begin{align*} v(t)&=\frac{x(t + \Delta t)-x(t)}{\Delta t}\ &=\frac{3(t+\Delta t)^{2}+4(t+\Delta t)-(3t^{2}+4t)}{\Delta t} \end{align*} $$

Step2: Expand the numerator

$$ \begin{align*} 3(t+\Delta t)^{2}+4(t+\Delta t)-(3t^{2}+4t)&=3(t^{2}+2t\Delta t+\Delta t^{2})+4t + 4\Delta t-3t^{2}-4t\ &=3t^{2}+6t\Delta t+3\Delta t^{2}+4t + 4\Delta t-3t^{2}-4t\ &=6t\Delta t+3\Delta t^{2}+4\Delta t \end{align*} $$

Step3: Simplify the velocity formula

$$ \begin{align*} v(t)&=\frac{6t\Delta t+3\Delta t^{2}+4\Delta t}{\Delta t}\ &=\frac{\Delta t(6t + 3\Delta t+4)}{\Delta t}\ &=6t + 3\Delta t+4 \end{align*} $$

Step4: Let (\Delta t\rightarrow0)

$$ v(t)=\lim_{\Delta t\rightarrow0}(6t + 3\Delta t+4)=6t+4 $$

Answer:

(v(t)=6t + 4)