The graph of f', the derivative of f, is the line shown in the figure below. If f(0)=5, then f(1)= ?

A) 0
B) 3
C) 6
D) 8
E) 11

Question

The graph of f', the derivative of f, is the line shown in the figure below. If f(0)=5, then f(1)= ?

A) 0
B) 3
C) 6
D) 8
E) 11

anonymous · Answer

I chose choice D. Am I correct?

anonymous · Answer

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anonymous · Answer

So $$
f'(x) = -6x+6
$$

anonymous · Answer

$$
\int_0^1 -6x+x\;dx = f(1) - 5
$$Then you solve for $f(1)$

anonymous · Answer

whoops, that one $x$ should be $6$

anonymous · Answer

But I thought $$f(1)=f(0)+\int\limits_{0}^{1}f'(x)dx$$

anonymous · Answer

the integral meaning area under the curve from 0 to 1 which is 3

anonymous · Answer

$$
\int _a^bf'(x)\;dx = f(b) - f(a)
$$

anonymous · Answer

$$
\int _0^1 f'(x) \;dx=f(1) - f(0)
$$$$
\int _0^1 (-6x+6)\;dx=f(1) - f(0)
$$$$
\int _0^1 (-6x+6)\;dx=f(1) - 5
$$

anonymous · Answer

Ok, but doing what you suggest gives me a negative number

anonymous · Answer

$$
-3x^2+6x\bigg|_0^1 = f(1)-5
$$$$
-3+6 +5 = f(1)
$$$$
8=f(1)
$$

anonymous · Answer

Oops, bad arithmetic mistake. Thanks, so now why is 5 being subtracted?

anonymous · Answer

Because $f(0)=5$, and $f(0)$ is being subtracted.