given the function below\n\n$f(x)=\\sqrt3{-80 x^{3}+144}$\n\nfind the equation of the tangent line to the…

given the function below\n\n$f(x)=\\sqrt3{-80 x^{3}+144}$\n\nfind the equation of the tangent line to the graph of the function at $x = 1$. answer in $m x+b$ form.\n\n$l(x)=$\n\nuse the tangent line to approximate $f(1.1)$.\n\n$l(1.1)=$\n\ncompute the actual value of $f(1.1)$. what is the error between the function value and the linear approximation?\nanswer as a positive value only.\n\n$|$error$| \\approx$ (approximate to at least 5 decimal places.)\n\nquestion help: video

given the function below\n\n$f(x)=\\sqrt3{-80 x^{3}+144}$\n\nfind the equation of the tangent line to the graph of the function at $x = 1$. answer in $m x+b$ form.\n\n$l(x)=$\n\nuse the tangent line to approximate $f(1.1)$.\n\n$l(1.1)=$\n\ncompute the actual value of $f(1.1)$. what is the error between the function value and the linear approximation?\nanswer as a positive value only.\n\n$|$error$| \\approx$ (approximate to at least 5 decimal places.)\n\nquestion help: video

Answer

Explanation:

Step1: 求(f(1))

将(x = 1)代入(f(x)=\sqrt[3]{-80x^{3}+144})，得(f(1)=\sqrt[3]{-80\times1^{3}+144}=\sqrt[3]{64}=4)。

Step2: 求(f^{\prime}(x))

根据复合函数求导法则((u^n)^\prime=nu^{n - 1}u^\prime)，令(u=-80x^{3}+144)，(n=\frac{1}{3})，则(f^{\prime}(x)=\frac{1}{3}(-80x^{3}+144)^{-\frac{2}{3}}\times(-240x^{2})=\frac{-80x^{2}}{(-80x^{3}+144)^{\frac{2}{3}}})。

Step3: 求(f^{\prime}(1))

将(x = 1)代入(f^{\prime}(x))，得(f^{\prime}(1)=\frac{-80\times1^{2}}{(-80\times1^{3}+144)^{\frac{2}{3}}}=\frac{-80}{64^{\frac{2}{3}}}=\frac{-80}{16}=-5)。

Step4: 求切线方程(L(x))

根据点斜式(y - y_0 = m(x - x_0))（这里(x_0 = 1)，(y_0 = 4)，(m=-5)），(L(x)-4=-5(x - 1))，整理得(L(x)=-5x + 9)。

Step5: 求(L(1.1))

将(x = 1.1)代入(L(x)=-5x + 9)，得(L(1.1)=-5\times1.1 + 9=3.5)。

Step6: 求(f(1.1))

(f(1.1)=\sqrt[3]{-80\times(1.1)^{3}+144}=\sqrt[3]{-80\times1.331 + 144}=\sqrt[3]{-106.48+144}=\sqrt[3]{37.52}\approx3.3476)。

Step7: 求(\vert error\vert)

(\vert error\vert=\vert f(1.1)-L(1.1)\vert=\vert3.3476 - 3.5\vert=0.1524)。

Answer:

(L(x)=-5x + 9)；(L(1.1)=3.5)；(\vert error\vert\approx0.1524)。