Exam 2 Review

Derivatives

100

g(x)=7/2x^2-3x+12

g'(x)=7/2x-3

200

s(t)=1/t+1/t^2

s'(t)=-1/t^2-2/t^3

300

g(x)=(x+2\sqrtx)e^x

g'(x)=\left(x + 2 \sqrt{x}\right) \mathrm{e}^{x} + \left(\frac{1}{\sqrt{x}} + 1\right) \mathrm{e}^{x}

Derivatives 2

100

y=(4x^2+3)(2x+5)

y'=2(4x^{2} + 3) + 8x(2x + 5)

200

y=sin(cos(x))

y'=-cos(cos(x))sin(x)

300

f(z)=e^{z/(z-1)}

f'(z)=(\frac{1}{z - 1} - \frac{z}{(z - 1)^{2}})e^{\frac{z}{z - 1}}

Derivatives 3

100

x^2-4xy+y^2=4

y'=\frac{2x-4y}{4x+2y}

200

\sqrt{x+y}=x^4+y^4

y'=\frac{8x^3-(x+y)^{-1/2}}{(x+y)^{-1/2}-8y^3}

300

Find the tangent line to  2(x^2+y^2)^2=25(x^2-y^2) at  (3,1) .

y=-9/13x+40/13

Applications

100

The area of a triangle with side lengths  a  and  b  and angle  theta  is given by  A=1/2ab sin \theta .
If  a=2 cm, b=3 cm, and  theta  increases at a rate of  0.2 rad/min, How fast is the area increasing when  \theta = \pi/3 ?

 \frac{dA}{dt}=0.3 cm^2/min

200

Linear approx of  f(x)=cos(2x)  at  a=\pi/6 

L(x) = \frac{1}{2} - \sqrt{3}(x - \frac{\pi}{6})

300

Use Linear approx to approx  1/4.002 

\frac{1}{4.002} \approx 0.249875