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# Criswell Calculus

### AP Calculus Learning to Take Derivatives by Rules (Chapter 3)

Definition: The Intermediate Value Theorem (IVT) If "f" is a continuous function on the closed interval [a,b] and K falls between f(a) and f(b), then there must exist at least one c on the open interval (a,b) where f(c) =k
Definition: Average Rate of Change As a formula: {F(b) –F(a)} / (b –a), which would be the same as the definition of slope m = (y2 –y1) / (x2 – x1).
Definition: Derivative of a function "Global Derivative" Lim h–>0 [ f (x+h) – f(x) ] / h Be able to extract from the notation, "What is the function?" and at "What x value?"
What does it mean to take the derivative of a function? Instantaneous Rate of Change. Slope of the tangent line for any point on the curve.
Relationship of Derivatives on Projectiles Displacement or Position Function: s(t), Velocity: s '(t) = v(t), Acceleration: s '' (t) = v ' (t) = a(t).
Derivative at a Point "Disposable Derivative". Lim x–>c [ f (x) – f(c) ] / (x – c) Be able to extract from the notation, "What is the function?" and at "What x value?"
Mean Value Theorem: Finding x -value where m-tangent = m-secant States that if a function f is continuous on the closed interval [a,b], and differentiable on the open interval (a,b), then there exists at least point c in the interval (a,b) such that f'(c) is equal to the average rate of change over [a,b].
Estimating the Instantaneous Rate of Change The best estimate from a table of values will come from using values that are a little under and a little over the desired target. Example f' (5) ≈ m sec =[f (5.1) – f (4.9) ] / (5.1 –4.9) even if f(5) is known it should NOT be used in the estimate!
Power rule for exponents If f(x) = a*x^n, then f ' (x) = n*a*x^(n-1) This concept can be extended independently across numerous terms in an equation .
Product Rule if the function can be split into a product such as y = u*v, then y ' = u*d(v) + v*d(u) Be prepared for a play on notation such as h(x)=f(x)*g(x) whereas h'(x) = f(x)* g '(x) + g(x)* f ' (x)
d(sin (x)) / dx cos (x)
d(cos (z)) / dz - sin (z)
d(tan (b)) / db sec^2(b)
d(csc (r)) / dr - csc(r)*cot(r)
d(sec (w)) / dw sec(w)*tan(w)
d(cot(g))/ dg - csc^2 (g)
Dx ( ln(x) ) 1/x
Dx ( e^x ) e^x
Understand that ln (x) and e^x are inverse functions. if y = ln | e^(3x-7) |, then an equivalent form y = 3x -7 if f(x) = e^(ln |2x+5|), then an equivalent form f(x) = 2x+5
Quotient Rule Generically d(high /low) = [low*d(high) - high*d(low) ] / low^2 Again be prepared for notation questions: if h(x) = f(x) / g(x) , then h '(x) = [g(x)*f '(x) - f(x)* g '(x)] / ( g(x) )^2
Be able to confirm Dx (tan(x)) and Dk (cot(k)) using the quotient rule. Example: if f (x) = tan(x) equivalent form f(x) = (sin(x) / cos(x)) so, f' (x) = [ cos(x)*d(sin(x)) - sin(x)*d(cos(x)) ] / (cos(x))^2 = [cos^2(x) + sin^2(x)]/ (cos(x))^2 =1 / (cos(x))^2 =sec^2 (x)
Mean Value Theorem (MVT) and Desmos When working with the MVT on edia if you run into a function that you are not 100% certain on how to differentiate, there is a calculus function option that will graph the derivative to a function on Desmos. This could be extremely helpful.
Created by: Troy.Criswell
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