Increasing and Decreasing functions using interval notation












0














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







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    0














    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







    share|cite|improve this question

























      0












      0








      0







      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







      share|cite|improve this question













      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






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      asked 2 days ago









      The StatisticianThe Statistician

      96111




      96111






















          3 Answers
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          active

          oldest

          votes


















          1














          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).






          share|cite|improve this answer





















          • 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?
            – The Statistician
            2 days ago










          • @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.
            – Joel Pereira
            yesterday





















          0














          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.)






          share|cite|improve this answer





























            0














            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






            share|cite|improve this answer








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              3 Answers
              3






              active

              oldest

              votes








              3 Answers
              3






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              1














              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).






              share|cite|improve this answer





















              • 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?
                – The Statistician
                2 days ago










              • @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.
                – Joel Pereira
                yesterday


















              1














              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).






              share|cite|improve this answer





















              • 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?
                – The Statistician
                2 days ago










              • @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.
                – Joel Pereira
                yesterday
















              1












              1








              1






              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).






              share|cite|improve this answer












              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).







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 2 days ago









              Joel PereiraJoel Pereira

              68819




              68819












              • 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?
                – The Statistician
                2 days ago










              • @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.
                – Joel Pereira
                yesterday




















              • 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?
                – The Statistician
                2 days ago










              • @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.
                – Joel Pereira
                yesterday


















              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?
              – The Statistician
              2 days ago




              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?
              – The Statistician
              2 days ago












              @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.
              – Joel Pereira
              yesterday






              @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.
              – Joel Pereira
              yesterday













              0














              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.)






              share|cite|improve this answer


























                0














                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.)






                share|cite|improve this answer
























                  0












                  0








                  0






                  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.)






                  share|cite|improve this answer












                  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.)







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 days ago









                  KM101KM101

                  5,5511423




                  5,5511423























                      0














                      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






                      share|cite|improve this answer








                      New contributor




                      Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.























                        0














                        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






                        share|cite|improve this answer








                        New contributor




                        Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                        Check out our Code of Conduct.





















                          0












                          0








                          0






                          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






                          share|cite|improve this answer








                          New contributor




                          Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                          Check out our Code of Conduct.









                          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







                          share|cite|improve this answer








                          New contributor




                          Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                          Check out our Code of Conduct.









                          share|cite|improve this answer



                          share|cite|improve this answer






                          New contributor




                          Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                          Check out our Code of Conduct.









                          answered 2 days ago









                          Any BanyAny Bany

                          31




                          31




                          New contributor




                          Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                          Check out our Code of Conduct.





                          New contributor





                          Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                          Check out our Code of Conduct.






                          Any Bany is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                          Check out our Code of Conduct.






























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