Chapter 5 Review- Brief Calc

BasicDerivatives

100

What rules, if any, would we use to find the derivative of y = x²

Power Rule.

200

What rules, if any, would we use to find the derivative of y = (x²)(ln x)

Product Rule primarily, but also logarithm and Power rules.

300

What rules, if any, would we use to find the derivative of y = (x²)/(e^x)

Quotient Rule Primarily, Power and Exponential rules

Alternatively, negative exponent rule would lead to Product Rule primarily, Power rule and Exponential (chain) rule.

400

What rules, if any, would we use to find the derivative of y = (y²)(ln x)

Implicit differentiation, Product rule, Logarithm rule.

500

What rules, if any, would we use to find the derivative of y = x²(x²+1)⁵

Most likely: Product rule with Power and Chain rules

Potentially: Logarithmic differentiation, but all of ours will explicitly say so.

Logarithmic Derivatives

100

Find the second derivative of..

y=ln(x)

y' = 1/x

y" = -x^-2 = -1 / x²

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?

Q(t)= Q₀e^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 25 flies initially and 400 flies after 4 days.

How fast is the population of fruit flies growing in exactly 4 days?

y = 25e^0.6931t is the model.

Approximately 6,930 flies per day.

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