2.1 – Rates of Change and Tangents to Curves. Function Review. 2.1 – Rates of Change and Tangents to Curves. Function Review. 2.1 – Rates of Change and Tangents to Curves. Function Review. Pascal’s Triangle. 2.1 – Rates of Change and Tangents to Curves. R ate of change:. - PowerPoint PPT Presentation
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2.1 – Rates of Change and Tangents to CurvesFunction Review
𝑓 (𝑥 )=2𝑥−7𝑓 (3 )=?
𝑓 (3 )=2 (3 )−7
𝑓 (3 )=6−7
𝑓 (3 )=−1
(3 ,−1 )
𝑓 (𝑥 )=2𝑥−7𝑓 (𝑎 )=?
𝑓 (𝑎 )=2 (𝑎 )−7
𝑓 (𝑎 )=2𝑎−7(𝑎 ,2𝑎−7 )
(𝑎 , 𝑓 (𝑎) )
𝑓 (𝑥 )=2𝑥−7𝑓 (3+𝑎)=?
𝑓 (3+𝑎)=2 (3+𝑎 )−7
𝑓 (3+𝑎)=6+2𝑎−7(3+𝑎 ,2𝑎−1 )
(3+𝑎 , 𝑓 (3+𝑎 ) )
2.1 – Rates of Change and Tangents to CurvesFunction Review
𝑓 (𝑥 )=𝑥2+2 𝑥−7𝑓 (𝑥+h )=?
𝑓 (𝑥+h )=(𝑥+h )2+2 (𝑥+h )−7
𝑓 (𝑥+h )=𝑥2+2 h𝑥 +h2+2 𝑥+2h−7
(𝑥+h , 𝑥2+2 h𝑥 +h2+2𝑥+2h−7 )
(𝑥+h , 𝑓 (𝑥+h ) )
2.1 – Rates of Change and Tangents to CurvesFunction Review
2.1 – Rates of Change and Tangents to CurvesInstantaneous Rate of Change or the Slope of a Tangent Line
𝑓 (𝑥 )=−2 𝑥2+4 (1,2 ) h
𝑚𝑡𝑎𝑛=𝑓 (1+h )− 𝑓 (1 )
(1+h )−1=¿
−2 (1+h )2+4− (−2 (1 )2+4 )h
=¿
−4h−2h2
h=¿
𝑚𝑡𝑎𝑛=−4−2h
−2 (1+2h+h2 )+4− (−2 (1 )2+4 )h
=¿
−2−4h−2h2+4−2h
=¿
h (−4−2h )h
=¿
𝑚𝑡𝑎𝑛=−4−2hh=0.1
h=0.01
𝑚𝑡𝑎𝑛=−4−2(0.1)𝑚𝑡𝑎𝑛=−4.2
𝑚𝑡𝑎𝑛=−4−2(0.01)𝑚𝑡𝑎𝑛=−4.02
h=0.001𝑚𝑡𝑎𝑛=−4−2(0.001)
𝑚𝑡𝑎𝑛=−4.002
2.2 – Limit of a Function and Limit Laws
Find the requested limits from the graph of the given function.
lim𝑥→ 0
𝑓 (𝑥 )=¿¿
lim𝑥→ 1
𝑓 (𝑥 )=¿¿
lim𝑥→−1
𝑓 (𝑥 )=¿¿
lim𝑥→ 2
𝑓 (𝑥 )=¿¿
lim𝑥→−2
𝑓 (𝑥 )=¿¿
−10
0
3
3
Defn: Limit
lim𝑥→𝑐
𝑓 (𝑥 )=𝐿
As the variable x approaches a certain value, the variable y approaches a certain value.
2.2 – Limit of a Function and Limit Laws
Given the following graph of a function, find the requested limit.
lim𝑥→ 2
𝑔 (𝑥 )=¿¿
𝑔 (2 )=¿
4
6
lim𝑥→𝑐
𝑓 (𝑥 )=𝐿
(2,6 )
2.2 – Limit of a Function and Limit Laws
Given the following graph of a function, find the requested limits.
lim𝑥→ 0
𝑓 (𝑥 )=¿¿ 𝑓 (0 )=¿
lim𝑥→−1
𝑓 (𝑥 )=¿¿
lim𝑥→1
𝑓 (𝑥 )=¿¿
lim𝑥→2
𝑓 (𝑥 )=¿¿
2 2
𝑗𝑢𝑚𝑝
2
𝑗𝑢𝑚𝑝
lim𝑥→𝑐
𝑓 (𝑥 )=𝐿
(0,2 )
𝑓 (−1 )=¿2(−1,2 )
𝑓 (1 )=¿3(1,3 )
𝑓 (2 )=¿3(2,3 )
2.2 – Limit of a Function and Limit LawsFind the requested limits for the given function.
lim𝑥→ 0
𝑓 (𝑥 )=¿¿
lim𝑥→1
𝑓 (𝑥 )=¿¿
lim𝑥→−1
𝑓 (𝑥 )=¿¿
lim𝑥→2
𝑓 (𝑥 )=¿¿
lim𝑥→−2
𝑓 (𝑥 )=¿¿
(0 )2−1=¿−1
(1 )2−1=¿0
(−1 )2−1=¿0
3(2 )2−1=¿
(−2 )2−1=¿3
𝑓 (𝑥 )=𝑥2−1
2.2 – Limit of a Function and Limit LawsA rational function is the ratio of two polynomial functions
𝑓 (𝑥 )= 𝑥+52 𝑥−4
lim𝑥→ 3
𝑥+52𝑥−4=¿ ¿
3+52(3)−4
=¿82=¿ 4
Substitution Theorem:If f(x) is a polynomial function or a rational function, then or .If f(x) is a rational function, then the denominator cannot equal zero.
2.2 – Limit of a Function and Limit Laws
2.2 – Limit of a Function and Limit Laws
𝑓 (𝑥 )=𝑘lim𝑥→𝑐
𝑘=𝑘
Constant Function
𝑓 (𝑥 )=𝑥lim𝑥→𝑐
𝑥=𝑐
Identity Function
Additional Limit Rules
2.2 – Limit of a Function and Limit Laws
2.2 – Limit of a Function and Limit Laws
lim𝑥→ 3
14−𝑥2
=¿¿
Find the following limits:
− 15
lim𝑥→ 5
𝑥𝑥2− 𝑥
=¿¿14
lim𝑥→−4
𝑥−6𝑥2−36
=¿¿ 12
lim𝑥→ 6
𝑥+2𝑥2+𝑥−10
=¿¿ 14
2.2 – Limit of a Function and Limit LawsFind the following limits: