What is the equation of the line tangent to f(x)=x2ex−xex2 at x=0?

Let's rewrite the function as f(x)=x2e−x−xe−x2Reminding that the Mac Laurin of ex is 1+x+x22+0(x2),the one of e−x is 1−x+x22+0(x2),and the one of e−x2 is 1−x2+x42+0(x4)By replacing them, we getf(x)=x2(1−x+x22+0(x2))−x(1−x2+x42+0(x4)).By carrying out the multiplication we get−x+x2+x42−x52+0(x4)In the end f'(0)=-1, i.e. the coefficient of the linear term in the Mac Laurin polynomial for f(x)

Question: "What is the equation of the line tangent..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Raven Gardner

Answered question

2022-04-08

What is the equation of the line tangent to

f (x) = \frac{x^{2}}{e^{x}} - \frac{x}{e^{x^{2}}}

at x=0?

Answer & Explanation

Wrastirtyzp9w

Beginner2022-04-09Added 10 answers

Let's rewrite the function as

f (x) = x^{2} e^{- x} - x e^{- x^{2}}

Reminding that the Mac Laurin of

e^{x}

1 + x + \frac{x^{2}}{2} + 0 (x^{2})

,
the one of

e^{- x}

1 - x + \frac{x^{2}}{2} + 0 (x^{2})

,
and the one of

e^{- x^{2}}

1 - x^{2} + \frac{x^{4}}{2} + 0 (x^{4})

By replacing them, we get

f (x) = x^{2} (1 - x + \frac{x^{2}}{2} + 0 (x^{2})) - x (1 - x^{2} + \frac{x^{4}}{2} + 0 (x^{4})) .

By carrying out the multiplication we get

- x + x^{2} + \frac{x^{4}}{2} - \frac{x^{5}}{2} + 0 (x^{4})

In the end f'(0)=-1, i.e. the coefficient of the linear term in the Mac Laurin polynomial for f(x)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?