ch. 6 review - campbell.k12.ky.us 6 review...ย ยท exponential applications the function ๐ฅ=100 .15...
TRANSCRIPT
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)โฆ including Separation of Variables Derivatives/Integrals of Transcendental
Functions (trig, exponential, logs)
Second Fundamental Theorem of Calculus
If f(x) = 2๐ฅcos ๐ก ๐๐ก, find fโ(x).
If g(x) = 15๐ฅ๐2๐ก ๐๐ก , find gโ(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 ๐ฅ
Simplifying Logs
2๐๐๐4๐ฅ
๐๐๐๐ฅ2
3 log 2
ln ๐ฅ2
๐ ๐๐๐ฅ
๐๐ฅ๐๐5
Derivatives of Logs/Logarithmic Differentiation
๐
๐๐ฅlog5 ๐ฅ ๐
๐๐ฅ7๐ฅ+2
๐
๐๐ฅ๐๐๐8(2๐ฅ โ 5) 12๐ฅ๐๐ฅ
๐
๐๐ฅ3๐ฅ5๐ฅ
Integration of Trig Functions
tan ๐ฅ ๐๐ฅ
cot ๐ฅ ๐๐ฅ
sec ๐ฅ ๐๐ฅ
csc ๐ฅ ๐๐ฅ
Trig Integrals
๐ ๐๐๐ฅ ๐๐ฅ = โ๐๐๐ ๐ฅ + ๐ ๐๐๐ ๐ฅ ๐๐ฅ = ๐ ๐๐๐ฅ + ๐
sec ๐ฅ ๐๐ฅ = ln | sec ๐ฅ + tan ๐ฅ| + ๐
csc ๐ฅ ๐๐ฅ = โln | csc ๐ฅ + cot ๐ฅ| + ๐
tan ๐ฅ ๐๐ฅ = ln | sec ๐ฅ | + ๐
cot ๐ฅ ๐๐ฅ = โln | csc ๐ฅ | + ๐
Integration
4
8๐ฅ โ 1๐๐ฅ
2๐2๐ฅ
5 โ 4๐2๐ฅ๐๐ฅ
5๐ฅ + 6
๐ฅ๐๐ฅ
2
(4๐ฅ โ 1)3๐๐ฅ
Integrate Trig Functions
tan(2๐ฅ + 5) ๐๐ฅ
sec 5๐ฅ
Separation of Variables
See Population Problem, pg. 269.
We now know how to solve this QUICKLY!!!
Exponential Applications
The function ๐ ๐ฅ = 100๐ .15๐ก 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) = ๐0๐
๐๐ก 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 fโ(x) if ๐ ๐ฅ =(3๐ฅ+7)5
3๐ฅ+2