How would you do these problems of derivatives and such?

Question

I know it's a lot of problems... but I am extremely confused... if anyone could even just pick and choose one and show me how they did it and what the answer was it would be so greatly appreciated!!! Thanks!!

1. For what values of x does the graph of f(x) = x^3 + 6x^2 + x + 7 have a horizontal tangent?

2. The equation of motion of a particle is s=2t^3-5t^2+8t+8, where s is in meters and t is in seconds.
(a) Find the velocity as a function of t.
(b) Find the acceleration as a function of t .
(c) Find the acceleration (in m/s^2) after 3s.

3. For f(x) = x + 3/x find f'(x).

4. Consider the function f(x) = 2e^x - 5x^5.
(a) Find the first derivative of the function.
(b) Find the second derivative of the function.


5. Differentiate the function r(t)= 4t^(-3/4)

6. Differentiate the function h(x) = (x-3)(6x-13)

7. Differentiate the function f(t)= 1/3t^9 - 6t^3 + 2t

Answer 1

All calculus at this is level is pattern recognition. I'll pick one problem to show you which patterns are used.



7. Differentiate the function f(t)= 1/3t^9 - 6t^3 + 2t



This uses the following patterns:



1. If f(x) = x^n, where n > 0, then f'(x) = nx^(n-1).



2. If f(x) = g(x) + h(x) then f'(x) = g'(x) + h'(x).



3. If f(x) = Ag(x), where A is a constant, then f'(x) = Ag'(x).



Now, notice that your problem function is a bunch of additions, which matches pattern 2. So now your problem is how to differentiate the following three functions:



1/3t^9

-6t^3

2t



Each of these functions matches pattern 3. So now your problem is how to differentiate the following three functions:



t^9

t^3

t



These all match pattern 1.



So the answer is:



3t^8 - 18t^2 + 2.



Eventually you get to the point where you visualize the exponent dropping down (pattern 1) to multiply with the constant (pattern 3), leaving a little of itself behind (continuing pattern 1), for each term in the equation (pattern 2).



Hope it helps. Don't get too bogged down in how complicated the whole function looks. Just split it up into smaller pieces and apply the patterns.

Answer 2

You have several derivatives to handle in the 7 problems.

1. This is a min or max problem. Take the derivative of f(x) and set it to zero. Its a quadratic, so you will have two roots. Solve for the 2 positions. Remember that d/dx of a linear term in x, such as (a x^n) is (a times n) x^(n-1) and the derivative of a constant is zero. This is the "power rule". The sign with "n" is important, as you will find out.



2. Take the derivative ds/dt to get the velocity function.

You get v= 6 t^2 - 10 t + 8

Take the derivative of the velocity function /dt to get the accleration function.

Evaluate the acceleration function for t=3



3. The second term is x^-1. Power rule holds.



4. Remember that d (e^x)/dx= e^x, the rest should be duck soup



5. the (-3/4) power follows the general power rule

6. Multiply and then take derivative

7. Piece of cake.

Answer 3

Couple of questions: a million) is h drawing close 0? 2) if h is drawing close 0, is it drawing close from the adverse or from the constructive? If from the neg., your answer would be neg.. If from the pos., your answer would be pos.. i'm not sure if the respond is 0, neg. infinity or pos. infinity. i could ought to plug in some very small numbers to make certain for specific. in no way ideas. i'm not sure if i replaced into any help. Sorry.