# Let f be a continuous function: a) Find #f(4) if ∫_0^(x^2) f(t) dt = x sin πx# for all #x#. b) Find #f(4) if ∫_0^ f(x)t^2 dt = x sin πx# for all #x#?

a)

a) Differentiate both sides.

Through the Second Fundamental Theorem of Calculus on the left-hand side and the product and chain rules on the right-hand side, we see that differentiation reveals that:

b) Integrate the interior term.

Evaluate.

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a) ( f(4) = 2 )

b) ( f(4) = \frac{1}{8} )

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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