Help plz!!!!!?!!!?!??!????

F(x) = e^-x . Find the equation of the tangent to f(x) at x=-1

natima [27]

Answer:

The equation of the tangent line is given by the following equation:

$\displaystyle y - \frac{1}{e} = \frac{-1}{e} \bigg( x - 1 \bigg)$

General Formulas and Concepts:

Algebra I

Point-Slope Form: y - y₁ = m(x - x₁)

x₁ - x coordinate
y₁ - y coordinate
m - slope

Calculus

Differentiation

Derivatives
Derivative Notation

Derivative Property [Multiplied Constant]:
$\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Rule [Basic Power Rule]:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
$\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

*Note:

Recall that the definition of the derivative is the slope of the tangent line.

Step 1: Define

Identify given.

$\displaystylef(x) = e^{-x} \\x = -1$

Step 2: Differentiate

[Function] Apply Exponential Differentiation [Derivative Rule - Chain Rule]:
$\displaystyle f'(x) = e^{-x}(-x)'$
[Derivative] Rewrite [Derivative Rule - Multiplied Constant]:
$\displaystyle f'(x) = -e^{-x}(x)'$
[Derivative] Apply Derivative Rule [Derivative Rule - Basic Power Rule]:
$\displaystyle f'(x) = -e^{-x}$

Step 3: Find Tangent Slope

[Derivative] Substitute in x = 1:
$\displaystyle f'(1) = -e^{-1}$
Rewrite:
$\displaystyle f'(1) = \frac{-1}{e}$

∴ the slope of the tangent line is equal to $\displaystyle \frac{-1}{e}$ .

Step 4: Find Equation

[Function] Substitute in x = 1:
$\displaystyle f(1) = e^{-1}$
Rewrite:
$\displaystyle f(1) = \frac{1}{e}$

∴ our point is equal to $\displaystyle \bigg( 1, \frac{1}{e} \bigg)$ .

Substituting in our variables we found into the point-slope form general equation, we get our final answer of:

$\displaystyle \boxed{ y - \frac{1}{e} = \frac{-1}{e} \bigg( x - 1 \bigg) }$

∴ we have our final answer.

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Learn more about derivatives: brainly.com/question/27163229

Learn more about calculus: brainly.com/question/23558817

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

3 0

2 years ago

Anna starts a fixed deposit with $178000, the bank offers 9.5% per year simple interest for 8 months. she has the option of rest

soldier1979 [14.2K]

Using simple interest, we have that:

A) The interest due after 8 months is $11,272.33.

B) The total value of the investment will be of $189,986.24.

The amount of interest earning using simple interest, after t years, with an yearly interest rate of i and an initial investment of P is given by:

$E = Pit$