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Calculus
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12 Nov 2019

1. Evaluate the limit (sin 2t)/ (sin 6t) as t -> 0

Limit=?

2. Evaluate the limit as x -> 0 sin(3x)/8x

Limit=?

3. Find limit as theta -> 0 (cos theta - 1)/(9 sin theta)

Limit=?

4. Differentiate h(t) = -2 csc t + e^t cot t

h'(t)=?

5. Let f(x) = (4tan(x))/x

f'(x)=?

f'(5)=?

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Elin Hessel
Elin HesselLv2
1 Mar 2019
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Related questions

1. Use logarithmic differentiation to find f '(t) when f(t)=1/[t(t+1)(t+2)

2. find the limit as theta approaches 0 of sin theta cot 2 theta

3. A model rocket shot straight up( purely vertical) from Earth's surface would reach a height above ground s after t seconds of s=960t-16t^2

-How high will the model rocket go?

4. Find dr/d theta when r=(2 +sec theta)sin theta

Please show work for solutions

Find the AntiDerivatives, use c for constant:

f(x) = (1/2)*x^2 - 2x + 3

f(x) = 7sqrt(x) + 3 cos x

Find f:

f ' (x) = sqrt(x) * (6+5x)

f(1) = 9

f''(theta) = sin (theta) + cos (theta), f(0) = 3, f'(0) = 2

f(theta) = ?

here is the problem it is already solved and this is the yahooanswer solution and my teachers solution as well. All i need is aGraph and for someone to explain how my teachergot 0,0 and 0,-2 as intresection points.

teachers solution:find all 4 points of intersections: (0,0),(0,-2), (-4sqrt(17sqrt(2)-24), 10sqrt(2)-14),(-4sqrt(17sqrt(2)-24), 10sqrt(2)-14).

yahoo answers:sin(theta) = r^2/4 ==> sin(theta)>= 0 . Hence 0 =< theta =< pi

r = 1 - sin(theta) ; therefore 0 =< r =< 1, because now weknow 0 =< sin(theta) =< 1

Therefore y = r sin(theta) >= 0 so the intersection points are in theupper half plane.

Also, sin(theta) = r^2/4 =< 1/4 < 1/sqrt(2) = sin pi/4 =sin(3pi/4),

so 0 =< theta < pi/4 or 3pi/4 < theta =< pi.

---
At an intersection point, sin(theta) = r^2/4 = 1 - r

r^2 + 4r - 4 = 0

Roots: r = -- 2 +/- 2 sqrt(2); but r >= 0, so r = 2 sqrt(2) - 2= 2(sqrt(2) - 1)

sin(theta) = 1 - r = 3 - 2 sqrt(2)

There is just one value of y, namely

y = r sin(theta) = 2 (sqrt(2) - 1)(3 - 2 sqrt(2) ) = 2 (5 sqrt(2) -7) = 10 sqrt(2) - 14

( or y = 10 (sqrt(2) - 1.4 ) = 0.142135...

However, there are two intersection points, with x^2 = r^2 -y^2

x = +/- sqrt( r^2 - y^2 )

x^2 = r^2 - y^2 = 4 (sin theta) -- y^2 = 4 (3 - 2 sqrt(2)) -- [2(5sqrt(2) - 7)]^2

x^2 = 12 - 8 sqrt(2) -4 [ 99 - 70 sqrt(2)] = 272 sqrt(2) - 384 (=17 x 2^4 sqrt(2) - 3 x 2^7)

x = +/- 4 sqrt(17 sqrt(2) - 24) [ = +/- 0.8161427...]

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