Limits
Evaluate lim π₯β2β‘ (8β3β’π₯+12β’π₯2), if it exists.
lim π₯β2 β‘(8β3β’π₯+12β’π₯2)= 8β3β’(2)+12β’(4)= 50
β«(4x7+5x3+7x+5)dx
=(1/2x)8+(5/4)x4+(7/2x)2+5x+C
Derivative of cos(x)
What is -sin(x)?
Differentiability
What is a function that is continuous and smooth with no sharp corners, holes, cusps, or vertical asymptotes?
Equation of a tangent line
(y - y1) = f'(a)(x - a)
L'Hopital's rule
What is a calculus method used to evaluate limits of fractions that result in indeterminate forms (0/0 or βΎοΈ/βΎοΈ) in which the limit of the quotient is equal to the limit of the quotient of their derivatives.
β«6x2cos(x3)dx
Let u=x3, since x3 is the inner function:
du/dx=3x2
du=3x2
The integral can then be simplified to:
2β«(cos(u))du =2sin(u)+C
Replacing u with x3 gives: 2sin(x3)+C
The quotient rule for [f(x)/g(x)]
What is [f'(x)g(x)-f(x)g'(x)]/[g(x)2]?
Critical points
x=c is a critical point if f(c) is defined and either f'(c) = 0 or f'(c) is undefined
What is the slope of the tangent line at x=2 for the function below.
f(x)=ln(2x)+x2
=9/2
Evaluate lim ββ0 β‘(6+β)2β36/β, if it exists
lim ββ0 β‘(6+β)2β36/β=
lim ββ0β‘ 36+12β’β+β2β36/β=
lim ββ0β‘ ββ‘(12+β)/β=
lim ββ0 β‘(12+β)=
12
β«ex+4+4dx
=ex+4+C
Derivative of cot(x)
What is -csc2(x)?
How to find relative maximums and relative minimums on graph f'(x)?
Relative maximum: Look for where f'(x) changes from positive to negative.
Relative minimums: Look for where f'(x) changes from negative to positive.
Find the average value of the function f(x)=sin(x)cos(x) on the interval [0,Ο/2].
=1/Ο
Requirements for a limit to exist (3)
1. Left-hand limit exists
2. Right-hand limit exists
3. Limits are equal
Solve d/dx(β«ln(t2+t)dt) using the bounds 0 and x2
ln((x2)2+x2)β 2x
=2xβ ln(x4+x2)
Find fβ²(x) of tan(sec(x))
Recall that the derivative of tan(u) is: sec2(u)β du
Then, in the given problem, the βuβ is sec(x). The derivative of u is then ddx(sec(x))=sec(x)tan(x). Then, by chain rule:
fβ²(x)= sec2(sec(x))β sec(x)tan(x)
Does f(x) have a relative maximum or minimum at a point where f'(x)=0 but the graph does not cross the x-axis?
No. If f'(x) does not change sign (e.g., remains positive), the function increases, pauses, and increases again.
Lucky Square! Share your favorite topic on calculus.
Good answer
Use the Squeeze Theorem to determine the value of lim π₯β0β‘ π₯4β’sinβ‘(π/π₯).
=0
Find the area bounded by the functions f(x)=x2+3x+2 and g(x)=x+5.
=32/3
Derivative of f(x)=ln(3x)
u=3x
f(x)=ln(u)
fβ²(x)=1/u(uβ²) =1/3x(3) =1x
The graph of f'(x) has a sharp corner (like a |x| shape) at x=2. Can a relative maximum occur at x=2? If so, why?
Yes, if f'(x) changes from positive to negative at x=2, a maximum exists even if the derivative is undefined at that point.
The fathers of calculus
Who is Sir Isaac Newton and Gottfried Wilhelm Leibniz?