Differentiate for the Rate, Integrate for the Total, and Read Slopes
Calculus in physics is a short list of rules plus one habit: the slope of a graph is the derivative and the area under it is the integral. Use those and the rules for a power, a product, sine and cosine.
Differentiation, Integration and Graph Slope is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
Rules you need
∫ dx/(a x + b) = ln(a x + b)/a.
The average value of a function over an interval is its integral divided by the interval length: for v = 2t + 3 between 0 and 1 s, the integral is 4, so the average velocity is 4 m/s.
Reading slopes on a graph
The slope of a graph at a point is the derivative there. A slope that keeps increasing means the curve bends upward (concave up); a local minimum is where the slope changes from negative to positive, and a local maximum where it changes from positive to negative. Compare slopes at several points by the steepness and sign of the tangent: m1 < m² < m3 means the curve gets steeper moving to the right.
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- ✓d(x^n)/dx = n x^(n-1); product rule for x sin x
- ✓∫ sin kx dx = -cos kx / k; ∫ cos²θ dθ = θ/2 + sin 2θ / 4; ∫ dx/(ax + b) = ln(ax + b)/a
- ✓a = dv/dt; average value = integral / interval length
- ✓Slope of a graph is the derivative; a minimum has slope changing from - to +
- ✓e = -dφ/dt: differentiate first, then substitute the time
Do not substitute the time before differentiating. For φ = 2t² - 3t + 4 at t = 2, differentiate first to get 4t - 3 and then substitute; putting t = 2 into φ and differentiating a constant gives zero.
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