At what points on the given curve x = 4t^3, y = 4 + 24t -...

At what points on the given curve $x = 4t^3$, $y = 4 + 24t - 14t^2$ does the tangent line have slope 1?\smaller x-value $(x, y) = ($\larger x-value $(x, y) = (rac{32}{27}, rac{124}{9})$

Transcript

00:01 So in this question we say at what points on the curve, x equals 4 t cubed, and y equals 4 plus 64 t minus 16 t squared, does the tangent line have a slope of 1? so if i want my tangeline to have a slope of 1, that means i want my d, y, d, d x to be equal to 1.

00:21 Now, what is dy, dx? you may recall that dydx is equal to the derivative of y with respect to, to t divided by the derivative of x with respect to t and i want this to be equal to one so my derivative of y with respect to t this time that's 64 minus 32 t over my d x d t this time that's 12 t squared i want this to be equal to one to solve this i'm going to multiply up my denominator multiply up that 12 t squared 12 t squared so that i have 64 minus 32t is equal to 12 t squared.

01:13 I'll set this equal to 0.

01:15 I'll say i've got 12 t squared plus 32 t minus 64 is equal to 0.

01:26 Now on that left -hand side, i'm going to factor out a greatest common factor.

01:31 I can factor out a 4.

01:34 If i divide both sides by four, i'm getting 3t squared plus 8t minus 16 is equal to zero.

01:46 Now at this point, i'm hoping that my left -hand side is going to factor.

01:52 And i believe this guy will factor.

01:55 Let's see how this is going to work.

01:57 So i believe i'm going to play around a little bit.

02:00 I think i've got 3t and t, and i think i have a 4.

02:05 And a 4.

02:07 I think i want, let's see, let's do t plus 4.

02:13 I think this will do it.

02:15 Because now i've got 12t minus 4t.

02:19 There's the 8t in the middle.

02:20 That's good...

Question

At what points on the given curve $x = 4t^3$, $y = 4 + 24t - 14t^2$ does the tangent line have slope 1?\smaller x-value $(x, y) = ($\larger x-value $(x, y) = ( rac{32}{27}, rac{124}{9})$

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At what points on the given curve $x = 4t^3$, $y = 4 + 24t - 14t^2$ does the tangent line have slope 1?\smaller x-value $(x, y) = ($\larger x-value $(x, y) = (rac{32}{27}, rac{124}{9})$