The left hand derivative of the given function comes out to be 3a² + 3ah + h².
Deducing the Left Derivative:
The given function is,
f(x) = x³ + 2
⇒ f(a) = a³ + 2
The left hand limit is the definition of the left-hand derivative of f: f′⁻(x) =
f(x+h)f(x)h. F is said to be left-hand differentiable at x if the left-hand derivative exists.
Now, the formula for the left derivative of a function is given as,
f'(a)⁻ = [ f(a+h) - f(a) ] / [ (a+h) - a]
f'(a)⁻ = [ ((a+h)³ + 2) - (a³+2) ] / h
f'(a)⁻ = (a³ + 3a²h + 3ah² + h³ + 2 - a³ - 2) / h
f'(a)⁻ = (3a²h + 3ah² + h³) / h
f'(a)⁻ = h(3a² + 3ah + h²) / h
f'(a)⁻ = 3a² + 3ah + h²
Hence, the left derivative is 3a² + 3ah + h².
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42 because 3:1 is 3 ounces for box a to 1 ounce for box b. So divide 128 by 3.
The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors
(x+4)(x+6)=0
x = -4, x=-6
The answer is C.