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.

Ella
  • 530
  • 4
  • 21

1 Answers1

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

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

user
  • 154,566
  • 1
    I got $y = e^1 x + 2$ for my answer for part c. Does this sound correct? – Ella Dec 03 '18 at 00:17
  • 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 $e^1$. – user Dec 03 '18 at 00:32