I've actually already done the first 2 problems I'm posting, but I'd like to see how someone else works them out because I'm not sure about my work. Also, I definitely need to see how to work out the last two for I'm also not sure about them. You'll definitely be repped (thanked).
1. Use the quotient rule to find the derivative of
You do not need to expand out your answer.
2. a)Find the derivative of: 6e-4xcos(9x). [Hint: use product rule and chain rule!]
b) find the equation of the tangent line to the curve at x=0. Write your answer in mx+b format.
3. a)Given the equation below, find dydx.
10x^10+8x30y+y^2=19b) find the equation of the tangent line to the curve at (1, 1). Write your answer in mx+b format
4. A fence 24 feet tall runs parallel to a tall building at a distance of 6 ft from the building.
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/ 24ft| | (bad pic, sorry)
We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.
[A] First, find a formula for the length of the ladder in terms of θ. (Hint: split the ladder into 2 parts.)
L(theta) = ?
[b] Now, find the derivative, L'(θ).
L'(theta) = ?
[C] Once you find the value of θ that makes L'(θ)=0, substitute that into your original function to find the length of the shortest ladder. (Give your answer accurate to 5 decimal places.)
L(theta(min)) is about ____?___ ft
10x^10 + 8x^30*y + y^2 = 19 ......differentiating implicity w.r.t x, we get:
100x^9 + 240x^29*y + 8x^30 dy/dx + 2y dy/dx = 0
=> 8x^30 dy/dx + 2y dy/dx = - 100x^9 - 240x^29*y
=> dy/dx(8x^30 + 2y) = - 100x^9 - 240x^29*y ...........factored out the dy/dx
=> dy/dx = (- 100x^9 - 240x^29*y)/(8x^30 + 2y)
so yeah, i got the same as you did (you should really use parenthesis to make what you are saying clear)
(b) at (1,1)
dy/dx = (- 100(1)^9 - 240(1)^29*(1))/(8(1)^30 + 2(1))
........= (-100 - 240)/(8 + 2)
can you take it from here?
For problems like these, ALWAYS DRAW A DIAGRAM. see the diagram below:
I labeled the appropriate angle t for theta. Note that we form two similar triagles, i called one A and one B. The hypotenuse of A is a and the hypotenuse of B is b. so we have the length of the ladder as a + b.
usinf trig ratios we will realize that:
in triagle A, the side opposite to t is 24, therefore we have:
sin(t) = 24/a
=> a = 24/sin(t)
in triangle B, the base is 6, therefore we have:
cos(t) = 6/b
=> b = 6/cos(t)
therefore, the length of the ladder is given by:
L(t) = a + b = 24/sin(t) + 6/cos(t)
i think you can take it from here