Question

find the equation of the line tangent to the graph of y at x = 1.5. enter the equation of the tangent line here (in terms of the variable x): y = you can view a streaming video of the solution of a version of this problem at m119tube solution. submit answer next item y = 4 * 5^x.

Explanation:

Step1: Find the derivative of the function

The function is $y = 4\cdot5^{x}$. Using the formula for the derivative of $a^{x}$ which is $a^{x}\ln(a)$, the derivative $y'=4\cdot5^{x}\ln(5)$.

Step2: Evaluate the derivative at $x = 1.5$

Substitute $x = 1.5$ into $y'$. $y'(1.5)=4\cdot5^{1.5}\ln(5)$. Calculate $5^{1.5}=\sqrt{5^{3}} = 5\sqrt{5}\approx5\times2.236 = 11.18$. Then $y'(1.5)=4\times11.18\times1.6094\approx72.07$.

Step3: Find the value of the function at $x = 1.5$

$y(1.5)=4\cdot5^{1.5}=4\times5\sqrt{5}\approx4\times11.18 = 44.72$.

Step4: Use the point - slope form of a line

The point - slope form is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(1.5,44.72)$ and $m = 72.07$. So $y-44.72 = 72.07(x - 1.5)$. Expand to get $y=72.07x-72.07\times1.5 + 44.72$, $y=72.07x-108.105 + 44.72$, $y=72.07x - 63.385$.

Answer:

$y = 72.07x-63.385$