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Calculus Review
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How do I know where f is increasing?
It is where f prime is positive. Find the derivative and set equal to zero. Use test points to find where f prime is positive or negative.
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How do I know where f has inflection points?
It is where f double prime equals zero or is undefined and the sign changes. The f prime function changes from increasing to decreasing or vise-versa. It is where f prime has maximums or minimums.
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How do I know if the particle is moving to the left?
It is where f prime is negative. Find where f prime equals zero. Then check test points on f prime.
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How do I know if the particle is speeding up or slowing down?
Find v(t) and a(t): If they have the same sign the particle is speeding up. If they have different signs the particle is slowing down.
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What is speed? It is the absolute value of velocity.
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What do I do if the problem says find the particular solution y = f(x)?
This is asking you to find the original function that represents f. You are probably doing a separable variable problem.
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What does ππ¦ ππ₯ mean? This is asking for the first derivative.
It could also be written π ππ₯
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What does π 2 π¦ ππ₯Β² mean? This is asking for the second derivative.
It could also be written π 2 π π₯ 2 .
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What is the limit definition of a derivative?
Or π β² π = lim π₯βπ π π₯ βπ(π) π₯βπ
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What are the limit rules?
Is it a hidden derivative? If itβs approaching infinity and itβs a polynomial over a polynomial then use the horizontal asymptote rules. Can you factor to simplify and just plug in the numbers. LβHopitalβs rule: do f(x) and g(x) both approach 0 or Β±β, then lim π₯βπ π(π₯) π(π₯) = lim π₯βπ πβ²(π₯) πβ²(π₯) .
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How do I prove a function is continuous?
Show the left hand limit and the right hand limit are equal and are equal to f(x).
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Day One Practice
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Differentiate: arctan 2x
2 1+4π₯Β²
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Differentiate: π π₯ π βπ₯
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Differentiate: 2 3π₯ 3Γ 2 3π₯ Γlnβ‘(2)
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π¦= 1 π₯ , Find ππ¦ ππ₯ β2π₯ π₯ β2
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π π₯ = π₯β1 π₯ , find π β² π₯ . π₯ π₯ 2 β8π₯+1
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sin 2π₯ + cos 3π₯ ππ₯= βcosβ‘(2π₯) 2 + sinβ‘(3π₯) 3 + C
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π₯ π₯ 2 β6 dx= 1 2 lnβ π₯ 2 β6β+πΆ
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π β²β² π₯ = π₯ 2 π₯β4 π₯β8 . Find the x-coordinate(s) for points of inflection on f.
x=4 and x=8
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cos(xy)=x, find ππ¦ ππ₯ β csc π₯π¦ βπ¦ π₯
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Find the slope of the line tangent to the curve y=arctan(3x) at x = 1 3 .
3 2
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Day Two Practice
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Use a right Riemann sum with the four subintervals indicated by the data in the table to approximate π π‘ ππ‘. t (minutes) 5 9 12 20 W(t) degrees F 54.0 58.2 63.1 68.1 70 (70) =
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Is the previous estimate an overestimate or an underestimate?
It is an overestimate, because the function is always increasing and the right Riemann sum would be above the curve.
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5 π₯ 2 β2π₯π¦=πππ π₯, find ππ¦ ππ₯ . ππ¦ ππ₯ = β sin π₯ +2π¦β10π₯ β2π₯
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ππ¦ ππ₯ = βπ₯ π¦ Find the solution y = f(x) to the given differential equation with the initial condition f(-1) = 2. π¦= 4 π₯ 2 +1
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ππ¦ ππ₯ = βπ₯ π¦ 2 2 Write an equation for the line tangent to the graph of f at x = -1 if f(-1)=2.
y β 2 = 2(x + 1)
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Day Three Practice
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sinβ‘(3π₯)ππ₯ = βcosβ‘(3π₯) 3 +πΆ
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lim π₯β0 2π₯ 4 +5 π₯ 3 5π₯ 4 + 3π₯ 3 5 3
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lim π₯β0 π πππ₯ π₯ = 1
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π π₯ = π₯ 2 β4π₯+3ππππ₯β€2 ππ₯+1ππππ₯>2 The function f is defined above
π π₯ = π₯ 2 β4π₯+3ππππ₯β€2 ππ₯+1ππππ₯>2 The function f is defined above. For what value of k, if any is f continuous at x = 2? k = -1
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X=0 Where are the minimums? X=-1.5 and x = 6
The function f given by π π₯ = 1 2 π₯ 4 β3 π₯ 3 β9 π₯ 2 has a relative maximum at x = ? X=0 Where are the minimums? X=-1.5 and x = 6
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What is the slope of the line tangent to the graph of π¦= π βπ₯ π₯+3 at x=1?
β 5 16π
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lim ββ0 π 3+β β π 3 β = π 3
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lim π₯ββ 4π₯ 6 β5π₯ 3π₯ 6 + 7π₯ 2 = 4 3
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ππ¦ ππ₯ 0 π₯ 2 sin 2π‘ ππ‘= 2π₯(π ππ 2 π₯ 2 )
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