What is k?

Calculus Level 3

For what value of k k does the sum of the gradients of the function f ( x ) = 2 x 2 + k x − 3 f(x)=2x^2+kx-3 , for integers 1 ≤ x ≤ 5 1\le x \le 5 , equal 25?


The answer is -7.

Charley Shi by Charley Shi , 19, New Zealand

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1 solution

Poca Poca
Jul 3, 2018

The derivative is f ′ ( x ) = 4 x + k f'(x)=4x+k . We can express the sum that we need in terms of k k :

∑ i = 1 5 f ′ ( i ) = ( 4 ⋅ 1 + k ) + . . . + ( 4 ⋅ 5 + k ) = 4 ( 1 + . . . + 5 ) + 5 k = 60 + 5 k \begin{aligned} \sum_{i=1}^5{f'(i)}&=(4\cdot 1 +k) + ... + (4\cdot 5 +k)\\ &= 4(1+...+5)+5k\\ &=60+5k \end{aligned}

We know that the sum is 25 25 , so 60 + 5 k = 25 ⟹ k = − 7 . 60+5k=25 \implies \boxed{k=-7}.

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