Mixing
Increasing and Decreasing functions using interval notation
$begingroup$
I have a question where I am asked to indicate in "interval notation" when a given function is increasing or decreasing.
I just don't understand what I'm meant to be doing, as I have been given an interval, plus if I try to find the x-values when I put my function $f'(x)>0$ and $f'(x)<0$, I always get no x values out as the x-values are all in the real numbers.
This is what I have been given...
For the function:
$$g(x)=e^t$$where $t=sin(x)$.
On the interval $[0, 4pi]$, indicate in interval notation when it is increasing and when it is decreasing.
How am I meant to do this question? Any help would be most appreciated.
Thanks.
calculus functions derivatives
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
$endgroup$
add a comment |
$begingroup$
I have a question where I am asked to indicate in "interval notation" when a given function is increasing or decreasing.
I just don't understand what I'm meant to be doing, as I have been given an interval, plus if I try to find the x-values when I put my function $f'(x)>0$ and $f'(x)<0$, I always get no x values out as the x-values are all in the real numbers.
This is what I have been given...
For the function:
$$g(x)=e^t$$where $t=sin(x)$.
On the interval $[0, 4pi]$, indicate in interval notation when it is increasing and when it is decreasing.
How am I meant to do this question? Any help would be most appreciated.
Thanks.
calculus functions derivatives
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
$endgroup$
add a comment |
$begingroup$
I have a question where I am asked to indicate in "interval notation" when a given function is increasing or decreasing.
I just don't understand what I'm meant to be doing, as I have been given an interval, plus if I try to find the x-values when I put my function $f'(x)>0$ and $f'(x)<0$, I always get no x values out as the x-values are all in the real numbers.
This is what I have been given...
For the function:
$$g(x)=e^t$$where $t=sin(x)$.
On the interval $[0, 4pi]$, indicate in interval notation when it is increasing and when it is decreasing.
How am I meant to do this question? Any help would be most appreciated.
Thanks.
calculus functions derivatives
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
$endgroup$
I have a question where I am asked to indicate in "interval notation" when a given function is increasing or decreasing.
I just don't understand what I'm meant to be doing, as I have been given an interval, plus if I try to find the x-values when I put my function $f'(x)>0$ and $f'(x)<0$, I always get no x values out as the x-values are all in the real numbers.
This is what I have been given...
For the function:
$$g(x)=e^t$$where $t=sin(x)$.
On the interval $[0, 4pi]$, indicate in interval notation when it is increasing and when it is decreasing.
How am I meant to do this question? Any help would be most appreciated.
Thanks.
calculus functions derivatives
calculus functions derivatives
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
asked Jan 5 at 15:39
The StatisticianThe Statistician
96111
96111
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
So this is a question about the sign of the derivative. Recall that if $f^{,prime} > $ 0, then f is increasing whereas if $f^{prime}$ $<$ 0, then f is decreasing. So the first step is to find f$^{,prime}$:
$$ f = e^{sin(x)} text{ on [0,4$pi$]} $$
$$ f^prime = cos(x)e^{sin(x)}$$
Now you first want to find the critical points where $f^prime$ = 0. In this case, this only occus when $cos(x)$ = 0 in [0,4$pi$], namely $left{frac{pi}{2},frac{3pi}{2},frac{5pi}{2},frac{7pi}{2}right}$.
Now you break up the interval using the critical points as endpoints of your partition. Then you take sample values from each partition and plug them into $f^prime$. The sign of the derivative will be the same for any value in a given partition.In this case, we need only check the sign of $cos(x)$ since $e^{sin(x)}$ is never 0.
The subintervals where $f^prime$ > 0 (resp. < 0 ) is where f is increasing (resp. decreasing).
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
$endgroup$
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
add a comment |
$begingroup$
Hint: Using the Chain Rule, you get
$$f’(x) = cos(x)e^{sin(x)}$$
Clearly $e^{sin(x)} > 0$ for all $x$, so the real question is about where $cos(x) > 0$ and $cos(x) < 0$. (Recall the unit circle and the quadrants.)
answered Jan 5 at 15:49
KM101KM101
5,8681523
$endgroup$
add a comment |
$begingroup$
Calculate the derivative of g,
$g'(x)=cos(x)e^{sin(x)}$,
now discuss the sign of g' ,
if g'(x) is positive then g is increasing,
if g' is negative then g is decreasing
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
So this is a question about the sign of the derivative. Recall that if $f^{,prime} > $ 0, then f is increasing whereas if $f^{prime}$ $<$ 0, then f is decreasing. So the first step is to find f$^{,prime}$:
$$ f = e^{sin(x)} text{ on [0,4$pi$]} $$
$$ f^prime = cos(x)e^{sin(x)}$$
Now you first want to find the critical points where $f^prime$ = 0. In this case, this only occus when $cos(x)$ = 0 in [0,4$pi$], namely $left{frac{pi}{2},frac{3pi}{2},frac{5pi}{2},frac{7pi}{2}right}$.
Now you break up the interval using the critical points as endpoints of your partition. Then you take sample values from each partition and plug them into $f^prime$. The sign of the derivative will be the same for any value in a given partition.In this case, we need only check the sign of $cos(x)$ since $e^{sin(x)}$ is never 0.
The subintervals where $f^prime$ > 0 (resp. < 0 ) is where f is increasing (resp. decreasing).
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
$endgroup$
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
add a comment |
$begingroup$
So this is a question about the sign of the derivative. Recall that if $f^{,prime} > $ 0, then f is increasing whereas if $f^{prime}$ $<$ 0, then f is decreasing. So the first step is to find f$^{,prime}$:
$$ f = e^{sin(x)} text{ on [0,4$pi$]} $$
$$ f^prime = cos(x)e^{sin(x)}$$
Now you first want to find the critical points where $f^prime$ = 0. In this case, this only occus when $cos(x)$ = 0 in [0,4$pi$], namely $left{frac{pi}{2},frac{3pi}{2},frac{5pi}{2},frac{7pi}{2}right}$.
Now you break up the interval using the critical points as endpoints of your partition. Then you take sample values from each partition and plug them into $f^prime$. The sign of the derivative will be the same for any value in a given partition.In this case, we need only check the sign of $cos(x)$ since $e^{sin(x)}$ is never 0.
The subintervals where $f^prime$ > 0 (resp. < 0 ) is where f is increasing (resp. decreasing).
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
$endgroup$
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
add a comment |
$begingroup$
So this is a question about the sign of the derivative. Recall that if $f^{,prime} > $ 0, then f is increasing whereas if $f^{prime}$ $<$ 0, then f is decreasing. So the first step is to find f$^{,prime}$:
$$ f = e^{sin(x)} text{ on [0,4$pi$]} $$
$$ f^prime = cos(x)e^{sin(x)}$$
Now you first want to find the critical points where $f^prime$ = 0. In this case, this only occus when $cos(x)$ = 0 in [0,4$pi$], namely $left{frac{pi}{2},frac{3pi}{2},frac{5pi}{2},frac{7pi}{2}right}$.
Now you break up the interval using the critical points as endpoints of your partition. Then you take sample values from each partition and plug them into $f^prime$. The sign of the derivative will be the same for any value in a given partition.In this case, we need only check the sign of $cos(x)$ since $e^{sin(x)}$ is never 0.
The subintervals where $f^prime$ > 0 (resp. < 0 ) is where f is increasing (resp. decreasing).
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
$endgroup$
So this is a question about the sign of the derivative. Recall that if $f^{,prime} > $ 0, then f is increasing whereas if $f^{prime}$ $<$ 0, then f is decreasing. So the first step is to find f$^{,prime}$:
$$ f = e^{sin(x)} text{ on [0,4$pi$]} $$
$$ f^prime = cos(x)e^{sin(x)}$$
Now you first want to find the critical points where $f^prime$ = 0. In this case, this only occus when $cos(x)$ = 0 in [0,4$pi$], namely $left{frac{pi}{2},frac{3pi}{2},frac{5pi}{2},frac{7pi}{2}right}$.
Now you break up the interval using the critical points as endpoints of your partition. Then you take sample values from each partition and plug them into $f^prime$. The sign of the derivative will be the same for any value in a given partition.In this case, we need only check the sign of $cos(x)$ since $e^{sin(x)}$ is never 0.
The subintervals where $f^prime$ > 0 (resp. < 0 ) is where f is increasing (resp. decreasing).
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
answered Jan 5 at 15:52
Joel PereiraJoel Pereira
74519
74519
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
add a comment |
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong?
$endgroup$
– The Statistician
Jan 5 at 17:05
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
$begingroup$
@TheStatistician Yes. when you pick a sample point $x^star$ in some subinterval S, and find the sign of $f^{,prime}(x^{star})$, then every point in S has the same sign.
$endgroup$
– Joel Pereira
Jan 6 at 15:58
add a comment |
$begingroup$
Hint: Using the Chain Rule, you get
$$f’(x) = cos(x)e^{sin(x)}$$
Clearly $e^{sin(x)} > 0$ for all $x$, so the real question is about where $cos(x) > 0$ and $cos(x) < 0$. (Recall the unit circle and the quadrants.)
answered Jan 5 at 15:49
KM101KM101
5,8681523
$endgroup$
add a comment |
$begingroup$
Hint: Using the Chain Rule, you get
$$f’(x) = cos(x)e^{sin(x)}$$
Clearly $e^{sin(x)} > 0$ for all $x$, so the real question is about where $cos(x) > 0$ and $cos(x) < 0$. (Recall the unit circle and the quadrants.)
answered Jan 5 at 15:49
KM101KM101
5,8681523
$endgroup$
add a comment |
$begingroup$
Hint: Using the Chain Rule, you get
$$f’(x) = cos(x)e^{sin(x)}$$
Clearly $e^{sin(x)} > 0$ for all $x$, so the real question is about where $cos(x) > 0$ and $cos(x) < 0$. (Recall the unit circle and the quadrants.)
answered Jan 5 at 15:49
KM101KM101
5,8681523
$endgroup$
Hint: Using the Chain Rule, you get
$$f’(x) = cos(x)e^{sin(x)}$$
Clearly $e^{sin(x)} > 0$ for all $x$, so the real question is about where $cos(x) > 0$ and $cos(x) < 0$. (Recall the unit circle and the quadrants.)
answered Jan 5 at 15:49
KM101KM101
5,8681523
answered Jan 5 at 15:49
KM101KM101
5,8681523
answered Jan 5 at 15:49
KM101KM101
5,8681523
answered Jan 5 at 15:49
KM101KM101
5,8681523
5,8681523
add a comment |
add a comment |
$begingroup$
Calculate the derivative of g,
$g'(x)=cos(x)e^{sin(x)}$,
now discuss the sign of g' ,
if g'(x) is positive then g is increasing,
if g' is negative then g is decreasing
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
$endgroup$
add a comment |
$begingroup$
Calculate the derivative of g,
$g'(x)=cos(x)e^{sin(x)}$,
now discuss the sign of g' ,
if g'(x) is positive then g is increasing,
if g' is negative then g is decreasing
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
$endgroup$
add a comment |
$begingroup$
Calculate the derivative of g,
$g'(x)=cos(x)e^{sin(x)}$,
now discuss the sign of g' ,
if g'(x) is positive then g is increasing,
if g' is negative then g is decreasing
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
$endgroup$
Calculate the derivative of g,
$g'(x)=cos(x)e^{sin(x)}$,
now discuss the sign of g' ,
if g'(x) is positive then g is increasing,
if g' is negative then g is decreasing
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
answered Jan 5 at 15:50
Pedro AlvarèsPedro Alvarès
484
484
add a comment |
add a comment |
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