Calculus.mathlife -

Pick a single problem type (e.g., finding velocity from position) and solve 5–10 practice problems. Then move to the next. Mastery comes from doing, not just reading.

[ f'(x) = \lim_h \to 0 \fracf(x+h) - f(x)h ]

| Integral ( \int f(x) , dx ) | Result (plus constant ( C )) | | :--- | :--- | | ( \int x^n , dx ) (n ≠ -1) | ( \fracx^n+1n+1 ) | | ( \int \frac1x , dx ) | ( \ln |x| ) | | ( \int e^x , dx ) | ( e^x ) | | ( \int \cos x , dx ) | ( \sin x ) | | ( \int \sin x , dx ) | ( -\cos x ) | This theorem connects the two pillars. It says: calculus.mathlife

| Function ( f(x) ) | Derivative ( f'(x) ) | | :--- | :--- | | Constant ( c ) | 0 | | ( x^n ) | ( n x^n-1 ) | | ( e^x ) | ( e^x ) | | ( \ln x ) | ( 1/x ) | | ( \sin x ) | ( \cos x ) | | ( \cos x ) | ( -\sin x ) | Core Question: What total amount builds up from a continuously changing rate?

Where ( F ) is any antiderivative of ( f ). Pick a single problem type (e

Differentiation and integration are inverse operations.

Interpretation: We slice the area under a curve into infinitely thin rectangles, sum them up, and get the exact total. [ f'(x) = \lim_h \to 0 \fracf(x+h) -

Meaning: If you integrate a function and then differentiate the result, you get back the original function.