Ch. 6 ReviewAP Calculus

Topics

6.2: Integrals of Reciprocal Functions 6.2: Second Fundamental Theorem of

Calculus 6.3: Log Properties (The Big Four) 6.4: Solving Exponential Equations (logs) 6.4: Logarithmic Differentiation

(exponential functions) Growth/Decay Problems (using logs to

solve)… including Separation of Variables Derivatives/Integrals of Transcendental

Functions (trig, exponential, logs)

Second Fundamental Theorem of Calculus

If f(x) = 2𝑥cos 𝑡 𝑑𝑡, find f’(x).

If g(x) = 15𝑥𝑒2𝑡 𝑑𝑡 , find g’(x).

Example 8, pg. 276 (or #58, pg. 278)

Differentiation/Integration Methods

Power Rule, Chain Rule

Product Rule, Quotient Rule

e^x 5^x ln x log3 𝑥

Simplifying Logs

2𝑒𝑙𝑛4𝑥

𝑙𝑛𝑒𝑥2

3 log 2

ln 𝑥2

𝑠𝑖𝑛𝑥

𝑒𝑥𝑙𝑛5

Derivatives of Logs/Logarithmic Differentiation

𝑑

𝑑𝑥log5 𝑥 𝑑

𝑑𝑥7𝑥+2

𝑑

𝑑𝑥𝑙𝑜𝑔8(2𝑥 − 5) 12𝑥𝑑𝑥

𝑑

𝑑𝑥3𝑥5𝑥

Integration of Trig Functions

tan 𝑥 𝑑𝑥

cot 𝑥 𝑑𝑥

sec 𝑥 𝑑𝑥

csc 𝑥 𝑑𝑥

Trig Integrals

𝑠𝑖𝑛𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐 𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐

sec 𝑥 𝑑𝑥 = ln | sec 𝑥 + tan 𝑥| + 𝑐

csc 𝑥 𝑑𝑥 = −ln | csc 𝑥 + cot 𝑥| + 𝑐

tan 𝑥 𝑑𝑥 = ln | sec 𝑥 | + 𝑐

cot 𝑥 𝑑𝑥 = −ln | csc 𝑥 | + 𝑐

Integration

4

8𝑥 − 1𝑑𝑥

2𝑒2𝑥

5 − 4𝑒2𝑥𝑑𝑥

5𝑥 + 6

𝑥𝑑𝑥

2

(4𝑥 − 1)3𝑑𝑥

Integrate Trig Functions

tan(2𝑥 + 5) 𝑑𝑥

sec 5𝑥

Separation of Variables

See Population Problem, pg. 269.

We now know how to solve this QUICKLY!!!

Exponential Applications

The function 𝑓 𝑥 = 100𝑒 .15𝑡 gives the size of

a rabbit population after t years.

a) How many rabbits are there after 10 years?

b) When does the population reach 1000?

c) What is the instantaneous rate of change of the population after 10 years? What are the units?

Exponential Growth/Decay

Know how to substitute given values into R(t) = 𝑎0𝑒

𝑘𝑡 formula.

Be able to recognize derivative (rate of change, instantaneous rate, slope of tangent, etc.) vs. integral (sum, area under curve, total accumulation).

Derivatives of Logs/Logarithmic Differentiation

Find f’(x) if 𝑓 𝑥 =(3𝑥+7)5

3𝑥+2