Considering the function $g(x) = 2 + e^x$.












4














Considering the function $g(x) = 2 + e^x$.



a) Find $g’(x)$.



So, this is simply derivative of the function. This would be $g’(x) = e^x$, right?



b) Explain how this shows that $g(x)$ is an increasing function for all values of $x$.



In this case, don’t we set the derivative to $0$ and find what the $x$ equals? Then, we put the $x$ values on a sign chart to find out if it is increasing or decreasing?



c) Find the equation of the tangent line to $g(x)$ at $x=1$.



For this part, we plug $x$ into our derivative to get the slope, right? Then we plug $x=1$ into the original function, $g(x)$ to get our $y$ value. Then find our $b$ value by plugging our y, x, and slope values.






functions







share|cite|improve this question


















  • 2




    b). One nice property of $exp$ is that it's always positive:$e^x > 0$ for all $x$.
    – Clement C.
    Dec 2 '18 at 23:37
















4














Considering the function $g(x) = 2 + e^x$.



a) Find $g’(x)$.



So, this is simply derivative of the function. This would be $g’(x) = e^x$, right?



b) Explain how this shows that $g(x)$ is an increasing function for all values of $x$.



In this case, don’t we set the derivative to $0$ and find what the $x$ equals? Then, we put the $x$ values on a sign chart to find out if it is increasing or decreasing?



c) Find the equation of the tangent line to $g(x)$ at $x=1$.



For this part, we plug $x$ into our derivative to get the slope, right? Then we plug $x=1$ into the original function, $g(x)$ to get our $y$ value. Then find our $b$ value by plugging our y, x, and slope values.






functions







share|cite|improve this question


















  • 2




    b). One nice property of $exp$ is that it's always positive:$e^x > 0$ for all $x$.
    – Clement C.
    Dec 2 '18 at 23:37














4












4








4







Considering the function $g(x) = 2 + e^x$.



a) Find $g’(x)$.



So, this is simply derivative of the function. This would be $g’(x) = e^x$, right?



b) Explain how this shows that $g(x)$ is an increasing function for all values of $x$.



In this case, don’t we set the derivative to $0$ and find what the $x$ equals? Then, we put the $x$ values on a sign chart to find out if it is increasing or decreasing?



c) Find the equation of the tangent line to $g(x)$ at $x=1$.



For this part, we plug $x$ into our derivative to get the slope, right? Then we plug $x=1$ into the original function, $g(x)$ to get our $y$ value. Then find our $b$ value by plugging our y, x, and slope values.






functions







share|cite|improve this question













Considering the function $g(x) = 2 + e^x$.



a) Find $g’(x)$.



So, this is simply derivative of the function. This would be $g’(x) = e^x$, right?



b) Explain how this shows that $g(x)$ is an increasing function for all values of $x$.



In this case, don’t we set the derivative to $0$ and find what the $x$ equals? Then, we put the $x$ values on a sign chart to find out if it is increasing or decreasing?



c) Find the equation of the tangent line to $g(x)$ at $x=1$.



For this part, we plug $x$ into our derivative to get the slope, right? Then we plug $x=1$ into the original function, $g(x)$ to get our $y$ value. Then find our $b$ value by plugging our y, x, and slope values.






functions




functions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 2 '18 at 23:35









Ella

32111




32111








  • 2




    b). One nice property of $exp$ is that it's always positive:$e^x > 0$ for all $x$.
    – Clement C.
    Dec 2 '18 at 23:37














  • 2




    b). One nice property of $exp$ is that it's always positive:$e^x > 0$ for all $x$.
    – Clement C.
    Dec 2 '18 at 23:37








2




2




b). One nice property of $exp$ is that it's always positive:$e^x > 0$ for all $x$.
– Clement C.
Dec 2 '18 at 23:37




b). One nice property of $exp$ is that it's always positive:$e^x > 0$ for all $x$.
– Clement C.
Dec 2 '18 at 23:37










1 Answer
1






active

oldest

votes


















2














For point a) that correct indeed



$$frac{d}{dx}(2+e^x)=0+e^x=e^x$$



As noticed by ClementC. in the comments, for b) recall that $forall x quad e^x>0$.



For point c), yes let consider $m=g'(1)$ and then recall that the line passing through $(x_0,y_0)$ is given by



$$y-y_0=m(x-x_0)$$






share|cite|improve this answer

















  • 1




    I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
    – Ella
    Dec 3 '18 at 0:17






  • 2




    Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
    – gimusi
    Dec 3 '18 at 0:32











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1 Answer
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1 Answer
1






active

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2














For point a) that correct indeed



$$frac{d}{dx}(2+e^x)=0+e^x=e^x$$



As noticed by ClementC. in the comments, for b) recall that $forall x quad e^x>0$.



For point c), yes let consider $m=g'(1)$ and then recall that the line passing through $(x_0,y_0)$ is given by



$$y-y_0=m(x-x_0)$$






share|cite|improve this answer

















  • 1




    I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
    – Ella
    Dec 3 '18 at 0:17






  • 2




    Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
    – gimusi
    Dec 3 '18 at 0:32
















2














For point a) that correct indeed



$$frac{d}{dx}(2+e^x)=0+e^x=e^x$$



As noticed by ClementC. in the comments, for b) recall that $forall x quad e^x>0$.



For point c), yes let consider $m=g'(1)$ and then recall that the line passing through $(x_0,y_0)$ is given by



$$y-y_0=m(x-x_0)$$






share|cite|improve this answer

















  • 1




    I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
    – Ella
    Dec 3 '18 at 0:17






  • 2




    Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
    – gimusi
    Dec 3 '18 at 0:32














2












2








2






For point a) that correct indeed



$$frac{d}{dx}(2+e^x)=0+e^x=e^x$$



As noticed by ClementC. in the comments, for b) recall that $forall x quad e^x>0$.



For point c), yes let consider $m=g'(1)$ and then recall that the line passing through $(x_0,y_0)$ is given by



$$y-y_0=m(x-x_0)$$






share|cite|improve this answer












For point a) that correct indeed



$$frac{d}{dx}(2+e^x)=0+e^x=e^x$$



As noticed by ClementC. in the comments, for b) recall that $forall x quad e^x>0$.



For point c), yes let consider $m=g'(1)$ and then recall that the line passing through $(x_0,y_0)$ is given by



$$y-y_0=m(x-x_0)$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 2 '18 at 23:39









gimusi

1




1








  • 1




    I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
    – Ella
    Dec 3 '18 at 0:17






  • 2




    Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
    – gimusi
    Dec 3 '18 at 0:32














  • 1




    I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
    – Ella
    Dec 3 '18 at 0:17






  • 2




    Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
    – gimusi
    Dec 3 '18 at 0:32








1




1




I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
– Ella
Dec 3 '18 at 0:17




I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct?
– Ella
Dec 3 '18 at 0:17




2




2




Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
– gimusi
Dec 3 '18 at 0:32




Yes that’s right, sorry at first I read a different thing. To check indeed both the function and the tangent pass through the point $(1,2+e)$ and the slope $g’(1)=e$ is fine. We don’t need to write $e^1$.
– gimusi
Dec 3 '18 at 0:32


















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