XP 3 points SCalcET8 2.8.502 Find the derivative of the function using the defin
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Question
XP 3 points SCalcET8 2.8.502 Find the derivative of the function using the definition of derivative. 6. 0 ft) = 4t-5t2 f'(t) State the domain of the function. (Enter your answer using interval notation.) State the domain of its derivative. (Enter your answer using interval notation.) 7.3 points SCalcET8 2.8.503 XP Find the derivative of the function using the definition of derivative. f(x) = 2.se-x + 3.7 f(x) = State the domain of the function. (Enter your answer in interval notation.) State the domain of its derivative. (Enter your answer in interval notation.)Explanation / Answer
Definition of derivative
f'(x) = lim h-->0 [f(x+h) - f(x)]/h
( 7 ) Given f (x ) = 2.5 x^2 - x + 3.7
f'(x)= lim h-->0 (2.5(x+h)^2 - (x+h) + 3.7 - 2.5x^2 + x - 3.7)/h
= lim h-->0 (2.5(x+h)^2 - x - h - 2.5x^2 + x)/h (cancelling the 3.7's)
= lim h-->0 (2.5(x+h)^2 - h - 2.5x^2)/h (cancelling the x's)
= lim h-->0 (2.5x^2 + 5xh - 2.5h^2 - h - 2.5x^2)/h (multiplying out 1.5(x+h)^2)
= lim h-->0 (5xh - 2.5h^2 - h)/h (cancelling out the 1.5x^2's)
= lim h-->0 (5x - 2.5h - 1)h/h (factoring out a common factor of h)
= lim h-->0 5x - 2.5h - 1 (h/h = 1)
= lim h-->0 (5x - 1) - lim h-->0 (2.5h) (the limit of a sum is the sum of the limits)
= 5x - 1 - 0 (taking the limits) so f'(x) => 5 x - 1
Domain
def : The domain of a function is the set of input or argument values for which the function is real and defined
f (x ) = 2.5 x^2 - x + 3.7
The function has no undefined points no domain constraints. Therefore, the domain is
- < x <
interval notation is ( - , )
Domain of derivative
f'(x) = 5 x - 1
The function has no undefined points no domain constraints. Therefore, the domain is
- < x <
interval notation is ( - , )
clearly, f'(x) also has a domain of the entire real numbers as well.
( 6 )
given f(t ) = 4 t - 5 t^2
f'(t) = lim h-->0 [f(t+h) - f(t)]/h
= lim h-->0 [4(t+h) - 5 (t+h)^2 - 4 t + 5 t^2 ]/h
= lim h-->0 [ 4t + 4h - 5 ( t^2 + 2th + h^2 ) - 4t + 5 t^2 ] /h
= lim h-->0 [ 4t + 4h - 5 t^2 - 10 th - 5 h^2 ) - 4t + 5 t^2 ] /h
= lim h-->0 [ 4h - 10 th - 5 h^2 ) ] /h
= lim h-->0 [ 4 - 10 t - 5 h ) - 4t ]h/h
= lim h-->0 [ 4 - 10 t - 5 h ) - 4t ]
= lim h-->0 (4 - 10 t) - lim h-->0 (5h)
= 4 - 10 t - 0
= 4 - 10 t
f'(t) = 4 - 10 t
f (x ) = 4 t - 5 t^2
The function has no undefined points no domain constraints. Therefore, the domain is
- < x <
interval notation is ( - , )
Domain of derivative
f'(x) = 4 - 10 t
The function has no undefined points no domain constraints. Therefore, the domain is
- < x <
interval notation is ( - , )
clearly, f'(x) also has a domain of the entire real numbers as well.
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