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This idea occurs in all basic physical science and engineering. In particular, for a projectile near the earth’s surface travelling straight up and down, ignoring air resistance, acted upon by no other forces but gravity, we have acceleration due to gravity = −32 feet/sec 2 Thus, letting s(t) be position at time t, we have s¨(t) = −32 We take this (approximate) physical fact as our starting point. From s¨ = −32 we integrate (or anti-differentiate) once to undo one of the derivatives, getting back to velocity: v(t) = s˙ = s(t) ˙ = −32t + vo where we are calling the constant of integration ‘vo ’.
At time t = 0 it has 10 bacteria in it, and at time t = 4 it has 2000. At what time will it have 100, 000 bacteria? Even though it is not explicitly demanded, we need to find the general formula for the number f (t) of bacteria at time t, set this expression equal to 100, 000, and solve for t. Again, we can take a little shortcut here since we know that c = f (0) and we are given that f (0) = 10. (This is easier than using the bulkier more general formula for finding c). And use the formula for k: k= 10 ln 2,000 ln 10 − ln 2, 000 ln 200 ln f (t1 ) − ln f (t2 ) = = = t1 − t2 0−4 −4 4 Therefore, we have f (t) = 10 · e ln 200 4 t = 10 · 200t/4 as the general formula.
3x2 + e2x − 11 + cos x dx =? sec2 x dx =? 7 1+x2 dx√=? 16x7 − x + √3x dx =? 2 23 sin x − √1−x dx =? 2 The simplest substitutions The simplest kind of chain rule application d f (ax + b) = a · f (x) dx (for constants a, b) can easily be run backwards to obtain the corresponding integral formulas: some and illustrative important examples are cos(ax + b) dx eax+b dx √ ax + b dx 1 ax+b dx = = = = 1 a · sin(ax + b) + C 1 ax+b +C a ·e 3/2 1 (ax+b) +C a · 3/2 1 · ln(ax + b) +C a Putting numbers in instead of letters, we have examples like cos(3x + 2) dx e4x+3 dx √ −5x + 1 dx 1 7x−2 dx = = = = 1 3 · sin(3x + 2) + C 1 4x+3 +C 4 ·e (−5x+1)3/2 1 +C −5 · 3/2 1 · ln(7x − 2) + C 7 Since this kind of substitution is pretty undramatic, and a person should be able to do such things by reflex rather than having to think about it very much.
Notes on first-year calculus by Garrett.