f(x)=x^3+3e^x

(a)
What is the derivative of f at x=0?

(b)
Using the (correct) value of f′(0), give the equation of the tangent line to the curve y=f(x) at the point (0,f(0))=(0,3). Your answer must be in the form of an equation for y in terms of x, that is, of the form y=L(x) for some linear function L(x).

Question

f(x)=x^3+3e^x

(a)	
What is the derivative of f at x=0? 

(b)	
Using the (correct) value of f′(0), give the equation of the tangent line to the curve y=f(x) at the point (0,f(0))=(0,3). Your answer must be in the form of an equation for y in terms of x, that is, of the form y=L(x) for some linear function L(x).

anonymous · Answer

@abb0t

amistre64 · Answer

fairly simple derivative, what do you come up with?

anonymous · Answer

3x^2 + 3e^x

amistre64 · Answer

good, and when x=0?

anonymous · Answer

undefine?

anonymous · Answer

cause x can never be 0

amistre64 · Answer

what property are you trying to distort to get that?

anonymous · Answer

3(x^2 +e^x) since e^x can't be 0 then it would be undefine right?

amistre64 · Answer

let x=0

3(0^2 + e^0)

we are replacing x with 0, not e^x with 0

anonymous · Answer

so the answer for a is 3?

amistre64 · Answer

yes, the derivative (the slope at x=0) is 3

amistre64 · Answer

now, when we know a slope and a point, how do we construct a line?/

anonymous · Answer

y=mx+b

amistre64 · Answer

thats slope and intercept .... which would work here if you recognize the given point (0,3) as a y intercept

amistre64 · Answer

if you can tell if its an intercept, we could just as well use the point slope format
$$y-y_o=m(x-x_0)$$

amistre64 · Answer

*if you cant tell ...

anonymous · Answer

i dunno then

amistre64 · Answer

if you dunno if its an intercept, lets use the point slope form to work with

anonymous · Answer

it's not an intercept that's for sure

amistre64 · Answer

for a slope of 3, and a point (0,3)

y - 3 = 3(x - 0)

now we can work it into a y=_______ construction

anonymous · Answer

y=3x+3

amistre64 · Answer

it may not be sure for you, but its pretty sure to me that it is a y intercept ..... but thats not important at the moment.  we simply need to use it in some useful fashion and a point slope format works either way

amistre64 · Answer

yep

anonymous · Answer

i just don't get the slope and the point part

amistre64 · Answer

what does the derivative of a function tell us about the function?

anonymous · Answer

the slope

amistre64 · Answer

and, they stated the point they wanted right?  (0,3)

amistre64 · Answer

so all it boils down to is using those parts to make the line equation for the tangent to the curve

anonymous · Answer

oh and then you just solve for yfinal okay i get it

amistre64 · Answer

practice makes it almost second nature

anonymous · Answer

okay thanks!