Chapter 4 Review- H. Calc

Trig. Derivatives

100

Find the first derivative of..

y=csc(x)

y'= -csc(x)cot(x)

200

Find the first derivative of..

y=cot(4x)

y'= -4csc²(4x)

300

Find the first derivative of..

y=4tan(x² +4)

y' = 8xsec²(x²+4)

400

Find the first derivative of..

y=x³cos(5x)

y' = 3x²cos(5x)-5x³sin(5x)

500

Find the first derivative of..

y=tan(x)sin(x)

y'  = sec²(x)sin(x)+tan(x)cos(x)

Logarithmic Derivatives

100

Find the first derivative of..

y=ln(x)

y' = 1/x

200

Find the first derivative of..

y=ln(5x³-12x)

y' = (15x²-12) / (5x³-12x)

300

Find the first derivative of..

y=2x⁴ln(x) 

y' = 8x³ln(x) + 2x³

400

Find the first derivative of..

y=ln(x²-1)³

y' = 6x / (x²-1)

500

Find the first derivative of..

y= ln[ x²(2x³-2x)²]

y' = 2/x  + (6x²-2)/(x³-x)

Misc.

100

Find the first derivative of..

y=e⁵

y'=0

200

Find the first derivative of..

y=e^-2x

y' = -2e^-2x

300

Find the first derivative of..

y=e^4xln(x)

y' = 4e^4xln(x) + (e^4x)/x

400

Find the first derivative of..

y=e⁴ln(2x⁵)

y' = 5e⁴ / x

500

Find the first derivative of..

y=-e^-4x / ln(2x)

y' = [4xln(2x)+1] / [xe^4xln(2x)²]

Exponential Derivatives

100

Find the first derivative of..

y= e^x

y'= e^x

200

Find the first derivative of..

y=4e^2x-2

y' = 8e^2x-2

300

Find the first derivative of..

y= e^-5x - e^x

y' = -5e^-5x - e^x

400

Find the first derivative of..

y=2x⁴e^-3x

y' = 8x³e^-3x - 6x⁴e^-3x

500

Find the first derivative of..

y= ln(e^x)

y' = 1

Growth & Decay

100

What is the general exponential growth and decay model?

y= Ce^kt

200

How would you know the difference between a growth and a decay model? Be specific.

Growth is an increase in the population, while decay is a decrease. Specifically, if the k value is positive, we have growth, but if the k value is negative we have decay. 

300

Nobelium has a half-life of 25 seconds. A sample contains 500 grams of Nobelium. How much Nobelium will remain after just 10 seconds?

y= 379.0272 grams

400

Find the exponential growth model for a population of fruit flies for which there are 100 flies after 2 days and 400 flies after 4 days.

y = 25e^0.6931t

500

Nobelium has a half-life of 25 seconds. A sample contains 500 grams of Nobelium. How much Nobelium will remain after 1 minute?    

y=94.8795 grams