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## Common Misconceptions (Calculus)

How does infinity really work? Is it the biggest number? Is it even a number at all?

# Calculus Misconceptions

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 $f'(x) = x\cdot 2^{x-1}.$

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