Determine the antiderivative F of “f” , which is defined by f (x) = 4x3 – 6, where F (0) = 3
Answers
Step-by-step explanation:
ɢⁱᵛᵉⁿ ᶠᵘⁿᶜᵗⁱᵒⁿ: ᶠ (ˣ) = 4ˣ3 – 6
ɴᵒʷ, ⁱⁿᵗᵉᵍʳᵃᵗᵉ ᵗʰᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ:
∫4ˣ3 – 6 = 4(ˣ4/4)-6ˣ + ᴄ
∫4ˣ3 – 6 = ˣ4 – 6ˣ + ᴄ
ᴛʰᵘˢ, ᵗʰᵉ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᵒᶠ ᵗʰᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ, ғ ⁱˢ ˣ4 – 6ˣ + ᴄ, ʷʰᵉʳᵉ ᴄ ⁱˢ ᵃ ᶜᵒⁿˢᵗᵃⁿᵗ
ᴀˡˢᵒ, ᵍⁱᵛᵉⁿ ᵗʰᵃᵗ, ғ(0) = 3,
ɴᵒʷ, ˢᵘᵇˢᵗⁱᵗᵘᵗᵉ ˣ = 0 ⁱⁿ ᵗʰᵉ ᵒᵇᵗᵃⁱⁿᵉᵈ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ, ʷᵉ ᵍᵉᵗ:
(0)4 – 6(0) + ᴄ = 3
ᴛʰᵉʳᵉᶠᵒʳᵉ, ᴄ = 3.
ɴᵒʷ, ˢᵘᵇˢᵗⁱᵗᵘᵗᵉ ᴄ = 3 ⁱⁿ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ
ʜᵉⁿᶜᵉ, ᵗʰᵉ ʳᵉᵠᵘⁱʳᵉᵈ ᵃⁿᵗⁱᵈᵉʳⁱᵛᵃᵗⁱᵛᵉ ᶠᵘⁿᶜᵗⁱᵒⁿ ⁱˢ ˣ4 – 6ˣ + 3
ʜᵒᵖᵉ ⁱᵗ'ˢ ʰᵉˡᵖ ᵘʰ ❤️
Given Function :ᶠ (ˣ) = 4ˣ3 – 6
ɴᵒʷ,integrate the function..
∫4ˣ3 – 6 = 4(ˣ4/4)-6ˣ + ᴄ
∫4ˣ3 – 6 = ˣ4 – 6ˣ + ᴄ
ᴛʰᵘˢ, the antiderivative function is the , ғ ⁱˢ ˣ4 – 6ˣ + ᴄ, ʷʰᵉʳᵉ ᴄ is a constant
ᴀˡˢᵒ, given that , ғ(0) = 3,
ɴᵒʷ, substitute ˣ = 0 in the antiderivative function ,we get
(0)4 – 6(0) + ᴄ = 3
ᴛʰᵉʳᵉᶠᵒʳᵉ, ᴄ = 3.
ɴᵒʷ, substitute ᴄ = 3 ⁱⁿ antiderivative function
Hence, the required antiderivative function isˣ4 – 6ˣ + 3