3.1 –Tangents and the Derivative at a Point

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Presentation transcript:

3.1 –Tangents and the Derivative at a Point Defn: Derivative: The slope of the tangent line to the graph of a function at a given point. The instantaneous rate of change of a function with respect to its variable. The limiting value of the ratio of the change in a function to the corresponding change in its independent variable. 𝑓 ′ 𝑥 𝑟𝑒𝑎𝑑𝑠,"𝑓 𝑝𝑟𝑖𝑚𝑒 𝑜𝑓 𝑥."

3.1 –Tangents and the Derivative at a Point

3.1 –Tangents and the Derivative at a Point Instantaneous Rate of Change / the Slope of a Tangent Line at a Point 𝑓 𝑥 =−2 𝑥 2 +4 1,2 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 ℎ −4−2ℎ ℎ 𝑚 𝑡𝑎𝑛 =𝑓′(𝑥)= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 𝑥+ℎ −𝑥 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −4−2ℎ 𝑚 𝑡𝑎𝑛 =𝑓′(1)= lim ℎ→0 𝑓 1+ℎ −𝑓 1 1+ℎ −1 = 𝑚 𝑡𝑎𝑛 =𝑓′(1)=−4 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −2 1+ℎ 2 +4− −2 1 2 +4 ℎ = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −2 1+2ℎ+ ℎ 2 +4− −2 1 2 +4 ℎ 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −2−4ℎ−2 ℎ 2 +4−2 ℎ 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim ℎ→0 −4ℎ−2 ℎ 2 ℎ

3.1 –Tangents and the Derivative at a Point Alternate Definition for the Derivative at a Point

3.1 –Tangents and the Derivative at a Point Instantaneous Rate of Change / the Slope of a Tangent Line at a Point 𝑓 𝑥 =−2 𝑥 2 +4 1,2 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥+1 𝑥−1 𝑥−1 𝑚 𝑡𝑎𝑛 =𝑓′(𝑎)= lim 𝑥→𝑎 𝑓 𝑥 −𝑓 𝑎 𝑥−𝑎 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥+1 𝑚 𝑡𝑎𝑛 =𝑓′(1)= lim 𝑥→1 𝑓 𝑥 −𝑓 1 𝑥−1 = 𝑚 𝑡𝑎𝑛 =𝑓′(1)=−4 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 +4− −2 1 2 +4 𝑥−1 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 +4−2 𝑥−1 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 +2 𝑥−1 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 1 = lim 𝑥→1 −2 𝑥 2 −1 𝑥−1

3.2 –The Derivative of a Function Differentiation is the process used to develop the derivative. Differentiating a function will create the derivative.

3.2 –The Derivative of a Function 𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 𝑓 𝑥 = 𝑥 2 +𝑥+1 𝑚 𝑡𝑎𝑛 =𝑓′(𝑥)= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 𝑥+ℎ 2 + 𝑥+ℎ +1− 𝑥 2 +𝑥+1 ℎ = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 𝑥 2 +2𝑥ℎ+ ℎ 2 +𝑥+ℎ+1− 𝑥 2 −𝑥−1 ℎ = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 2𝑥ℎ+ ℎ 2 +ℎ ℎ = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 ℎ 2𝑥+ℎ+1 ℎ = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim ℎ→0 2𝑥+ℎ+1 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 =2𝑥+1

3.2 –The Derivative of a Function 𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 𝑓 𝑥 = 𝑥 2 +𝑥+1 𝑚 𝑡𝑎𝑛 =𝑓′(𝑥)= lim 𝑎→𝑥 𝑓 𝑎 −𝑓 𝑥 𝑎−𝑥 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎+𝑥+1 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 +𝑎+1− 𝑥 2 +𝑥+1 𝑎−𝑥 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 =𝑥+𝑥+1 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 +𝑎+1− 𝑥 2 −𝑥−1 𝑎−𝑥 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 =2𝑥+1 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 +𝑎− 𝑥 2 −𝑥 𝑎−𝑥 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎 2 − 𝑥 2 +𝑎−𝑥 𝑎−𝑥 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎+𝑥 𝑎−𝑥 + 𝑎−𝑥 𝑎−𝑥 = 𝑚 𝑡𝑎𝑛 = 𝑓 ′ 𝑥 = lim 𝑎→𝑥 𝑎−𝑥 𝑎+𝑥+1 𝑎−𝑥