3.3c differentiation rule\n(6 points)\nif ( g(t)=-2 t^{4}+4 t^{2}+9 ) find\n( g(0)=)\n( g^{prime}(0)=)\n(…

3.3c differentiation rule\n(6 points)\nif ( g(t)=-2 t^{4}+4 t^{2}+9 ) find\n( g(0)=)\n( g^{prime}(0)=)\n( g^{prime prime}(0)=)\n( g^{prime prime prime}(0)=)\n( g^{(4)}(0)=)\n( g^{(5)}(0)=)\nnote: you can earn partial credit on this problem.\nnote: you are in the reduced scoring period. all work coun\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have 5 attempts remaining.

3.3c differentiation rule\n(6 points)\nif ( g(t)=-2 t^{4}+4 t^{2}+9 ) find\n( g(0)=)\n( g^{prime}(0)=)\n( g^{prime prime}(0)=)\n( g^{prime prime prime}(0)=)\n( g^{(4)}(0)=)\n( g^{(5)}(0)=)\nnote: you can earn partial credit on this problem.\nnote: you are in the reduced scoring period. all work coun\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have 5 attempts remaining.

Answer

Explanation:

Step1: Calculate (g(0))

Substitute (t = 0) into (g(t)=-2t^{4}+4t^{2}+9). (g(0)=-2\times0^{4}+4\times0^{2}+9 = 9)

Step2: Find the first - derivative (g^{\prime}(t))

Using the power rule ((x^{n})^\prime=nx^{n - 1}), (g^{\prime}(t)=(-2t^{4}+4t^{2}+9)^\prime=-2\times4t^{3}+4\times2t+0=-8t^{3}+8t) Then substitute (t = 0) into (g^{\prime}(t)), (g^{\prime}(0)=-8\times0^{3}+8\times0 = 0)

Step3: Find the second - derivative (g^{\prime\prime}(t))

Differentiate (g^{\prime}(t)=-8t^{3}+8t) using the power rule. (g^{\prime\prime}(t)=(-8t^{3}+8t)^\prime=-8\times3t^{2}+8=-24t^{2}+8) Substitute (t = 0) into (g^{\prime\prime}(t)), (g^{\prime\prime}(0)=-24\times0^{2}+8 = 8)

Step4: Find the third - derivative (g^{\prime\prime\prime}(t))

Differentiate (g^{\prime\prime}(t)=-24t^{2}+8) using the power rule. (g^{\prime\prime\prime}(t)=(-24t^{2}+8)^\prime=-24\times2t=-48t) Substitute (t = 0) into (g^{\prime\prime\prime}(t)), (g^{\prime\prime\prime}(0)=-48\times0 = 0)

Step5: Find the fourth - derivative (g^{(4)}(t))

Differentiate (g^{\prime\prime\prime}(t)=-48t) using the power rule. (g^{(4)}(t)=(-48t)^\prime=-48) Since (g^{(4)}(t)) is a constant, (g^{(4)}(0)=-48)

Step6: Find the fifth - derivative (g^{(5)}(t))

Differentiate (g^{(4)}(t)=-48) (a constant). The derivative of a constant is (0), so (g^{(5)}(t)=0) and (g^{(5)}(0)=0)

Answer:

(g(0)=9), (g^{\prime}(0)=0), (g^{\prime\prime}(0)=8), (g^{\prime\prime\prime}(0)=0), (g^{(4)}(0)=-48), (g^{(5)}(0)=0)