Calculus

Derivatives

Derivatives Warmup

         

Let f(x)=x3+x2+x+1.f(x) = x^3 + x^2 + x + 1.

What is the value of f′(1)?f'(1)?

by Brilliant Staff

Let nn be a real number and f(x)=xn+x−n.f(x) = x^n + x^{-n}.

What is the value of f′(1)?f'(1)?

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Cameron is out running in a straight path for an hour. His distance (in kilometers) from home tt hours after leaving is given by the formula s(t)=12(t−t2), for 0≤t≤1.s(t) = 12(t - t^2),\text{ for } 0 \leq t \leq 1.

His average velocity for the hour-long run is 0 km./h., since

s(1)−s(0)1−0=0−01−0=0.\frac{s(1) - s(0)}{1-0} = \frac{0-0}{1-0} = 0.

What is Cameron's average velocity (in km./h.) during the first half-hour of his trip?

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True or False?

For every quadratic function f(x)=ax2+bx+c,f(x) = ax^2 + bx + c, there is a real number xx such that f(x)=f′(x).f(x)=f'(x).

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True or False?

If nn is an odd integer and f(x)=xn,f(x) = x^n, then f′(x)f'(x) is an even function, i.e. f′(x)=f′(−x)f'(x) = f'(-x) for all xx in the domain of f′. f' .

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