Calculus
Common Misconceptions (Calculus)
If
\[f(x) = e^{x^2},\]
what is the correct expression for \(f'(x)?\)
Which statement(s) are true?
A. If \(f(x) = x^2,\) then \(f'(x) = 2x.\)
B. If \(f'(x) = 2x,\) then \(f(x) = x^2.\)
True or False?
Suppose \(f\) is a differentiable function on the interval \((a,b).\) If \(f\) is decreasing on \((a,b),\) then \(f'<0\) on \((a,b).\)
True or False?
\[\displaystyle \int_{-1}^1 \frac{1}{x^2} \,dx = \left . \dfrac{x^{-2+1}}{-2+1} \right |_{-1}^1 = -1 -1 = -2. \]
True or False?
If \[f(x) = 2^x,\] then \[f'(x) = x\cdot 2^{x-1}.\]