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III- (9 points)
Part A
In the plane referred to an orthopormal system, given the curve
(C^(')) representing the derivative function
f^(') of a function
f differentiable over
R.

Using the curve
(C), determine with justification:
a) The sense of variations of the function
f over
R.
b) The convexity of the function
f over
R.
Suppose that the function
f is defined over
R by
f(x)=(x+a)e^(-2) where
a is a real number.
a) Express
f^(')(x), the derivative function of
f over
R as a function of
a.
b) Determine graphically
f^(')(0) and then deduce the value of
a.

III- ( $9$ points) $\newline$ Part A $\newline$ In the plane referred to an orthopormal system, given the curve $\left(C^{\prime}\right)$ representing the derivative function $f^{\prime}$ of a function $f$ differentiable over $\mathbb{R}$ . $\newline$ $1$ ) Using the curve $(C)$ , determine with justification: $\newline$ a) The sense of variations of the function $f$ over $\mathbb{R}$ . $\newline$ b) The convexity of the function $f$ over $\mathbb{R}$ . $\newline$ $2$ ) Suppose that the function $f$ is defined over $\mathbb{R}$ by $f^{\prime}$ $1$ where $f^{\prime}$ $2$ is a real number. $\newline$ a) Express $f^{\prime}$ $3$ , the derivative function of $f$ over $\mathbb{R}$ as a function of $f^{\prime}$ $2$ . $\newline$ b) Determine graphically $f^{\prime}$ $7$ and then deduce the value of $f^{\prime}$ $2$ .

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Math Problems

Calculus

Find derivatives of sine and cosine functions

Full solution

Q. III- (

9

points)

\newline

Part A

\newline

In the plane referred to an orthopormal system, given the curve

\left(C^{\prime}\right)

representing the derivative function

f^{\prime}

of a function

f

differentiable over

\mathbb{R}

\newline

1

) Using the curve

(C)

, determine with justification:

\newline

a) The sense of variations of the function

f

over

\mathbb{R}

\newline

b) The convexity of the function

f

over

\mathbb{R}

\newline

2

) Suppose that the function

f

is defined over

\mathbb{R}

f^{\prime}

1

where

f^{\prime}

2

is a real number.

\newline

a) Express

f^{\prime}

3

, the derivative function of

f

over

\mathbb{R}

as a function of

f^{\prime}

2

\newline

b) Determine graphically

f^{\prime}

7

and then deduce the value of

f^{\prime}

2

Find Sign of $f'(x)$ : To find the sense of variations of $f$ , look at the sign of $f'(x)$ on the curve $(C)$ .
Determine Convexity of $f$ : If $f'(x) > 0$ , $f$ is increasing; if $f'(x) < 0$ , $f$ is decreasing.
Calculate $f'(x)$ : To determine the convexity of $f$ , look at the sign of $f''(x)$ , the second derivative of $f$ .
Find $f'(0)$ : If $f''(x) > 0$ , $f$ is convex; if $f''(x) < 0$ , $f$ is concave.
Graphical Interpretation: Now, let's find $f'(x)$ for $f(x) = (x + a)e^{-2}$ . $f'(x) = \frac{d}{dx}[(x + a)e^{-2}] = e^{-2} \cdot \frac{d}{dx}[x + a]$ since $e^{-2}$ is a constant.
Solve for $a$ : $f'(x) = e^{-2} \times (1) = e^{-2}$ since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Solve for $a$ : $f'(x) = e^{-2} \times (1) = e^{-2}$ since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .To find $f'(0)$ , plug in $x = 0$ into $f'(x) = e^{-2}$ . $\newline$ $f'(0) = e^{-2} \times (0 + a) = a \times e^{-2}$ .
Solve for $a$ : $f'(x) = e^{-2} \cdot (1) = e^{-2}$ since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .To find $f'(0)$ , plug in $x = 0$ into $f'(x) = e^{-2}$ . $\newline$ $f'(0) = e^{-2} \cdot (0 + a) = a \cdot e^{-2}$ .Using the graph of $f'(x)$ , determine the value of $f'(0)$ .
Solve for $a$ : $f'(x) = e^{-2} \cdot (1) = e^{-2}$ since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .To find $f'(0)$ , plug in $x = 0$ into $f'(x) = e^{-2}$ . $\newline$ $f'(0) = e^{-2} \cdot (0 + a) = a \cdot e^{-2}$ .Using the graph of $f'(x)$ , determine the value of $f'(0)$ .Graphically, if $f'(0)$ is given, then $f'(x) = e^{-2} \cdot (1) = e^{-2}$ $2$ equals that value.
Solve for a: $f'(x) = e^{-2} \times (1) = e^{-2}$ since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .To find $f'(0)$ , plug in $x = 0$ into $f'(x) = e^{-2}$ . $f'(0) = e^{-2} \times (0 + a) = a \times e^{-2}$ .Using the graph of $f'(x)$ , determine the value of $f'(0)$ .Graphically, if $f'(0)$ is given, then $x$ $1$ equals that value.Solve for $x$ $2$ by dividing the graphical value of $f'(0)$ by $x$ $4$ .