Calculus Basic Flashcards
Define the limit of f(x) as x approaches a.
The value L that f(x) gets arbitrarily close to when x is near a.
Evaluate lim(x→0) (sin x)/x.
1.
Evaluate lim(x→∞) 1/x.
0.
Evaluate lim(x→2) (x²−4)/(x−2) by factoring or direct substitution.
Use factoring: (x−2)(x+2)/(x−2)=x+2 → at x=2, result=4.
True/False: If f is continuous at x=a, then lim(x→a) f(x)=f(a).
True.
Find lim(x→∞) (2x²)/(x²+1).
Divide top/bottom by x² → limit=2/(1+0)=2.
lim(x→∞) (3x)/(x+4) in simplest form?
Divide by x: 3/(1+4/x)→3.
If f(x) has a removable discontinuity at x=a, how can it be fixed?
Redefine f(a)=lim(x→a) f(x) if that limit exists.
What does it mean if lim(x→a⁻) f(x)≠ lim(x→a⁺) f(x)?
f has a jump discontinuity at x=a (left/right limits differ).
State the Squeeze Theorem in short.
If g(x) ≤ f(x) ≤ h(x) and lim g(x)=lim h(x)=L, then lim f(x)=L.
Evaluate lim(x→∞) (5x+1)/(10x+3).
Divide top & bottom by x → (5+1/x)/(10+3/x)→5/10=1/2.
Evaluate lim(x→0) (1−cos x)/x² using known expansions.
Result=1/2 (using series or half-angle identity).
When is f(x) continuous at x=a?
If lim(x→a) f(x) exists, equals f(a), and f(a) is defined.
Find lim(x→∞) e^(−x).
0 (exponential decay).
Check continuity: f(x)=1/(x−1). Is it continuous at x=1?
No, it has a vertical asymptote at x=1.
lim(x→∞) (x²)/(e^x) → does it go to 0 or ∞?
0, because e^x grows faster than any polynomial.
Evaluate lim(x→π) (sin x)/(x−π) using L’Hôpital’s Rule or expansion.
Use L’Hôpital: derivative top cos x=−1, bottom=1 → limit=−1.
State L’Hôpital’s Rule condition in brief.
If limit is 0/0 or ∞/∞, we can take derivatives top and bottom.
If lim(x→2) f(x)=5, what is lim(x→2) [f(x)+3]?
5+3=8.
Define derivative of f at x=a: f’(a).
f’(a)=lim(h→0) [f(a+h)−f(a)]/h.
Derivative of f(x)=c (constant)?
0.
Derivative of f(x)=x^n?
nx^(n−1).
Derivative of f(x)=sin x?
cos x.
Derivative of f(x)=cos x?
−sin x.
Derivative of f(x)=e^x?
e^x.
Derivative of f(x)=ln x (x>0)?
1/x.
Derivative of f(x)=a^x (a>0)?
a^x ln a.
Product Rule formula for (u·v)’?
u’v + uv’.
Quotient Rule formula for (u/v)’?
[u’v − uv’] / v².
Chain Rule formula for d/dx [f(g(x))]?
f’(g(x)) · g’(x).
Derivative of f(x)=x^2+3x using power rule?
2x+3.
Derivative of f(x)=5x−7?
5.
Derivative of f(x)=√x = x^(1/2)?
(1/2)x^(−1/2).
Derivative of f(x)=1/x?
−1/x².
Find f’(x) if f(x)=2e^x + 3cos x.
2e^x − 3sin x.
Find f’(x) if f(x)=sin(3x) using chain rule.
3cos(3x).
Derivative of f(x)=ln(5x)?
1/(5x) · 5=1/x.
Derivative of f(x)= (x^3−1)/(x+1) using quotient rule or simplify first?
Simplify or quotient rule. If you do quotient rule, or factor.
Derivative of f(x)=e^(x^2)?
e^(x^2)·2x (chain rule).
Define critical point of f(x).
Where f’(x)=0 or f’(x) is undefined, but f(x) is defined.
If f’(a)=0, does it guarantee a local max or min?
Not necessarily; need further tests.
Second derivative test: if f’‘(a)>0 and f’(a)=0, what is f’s behavior at x=a?
Local minimum.
If f’‘(a)<0 and f’(a)=0, what is the local behavior at x=a?
Local maximum.
If f’‘(a)=0, does that guarantee an inflection point at x=a?
Not always, must check sign change in f’’.
Slope of tangent line to y=f(x) at x=a is given by?
f’(a).
Equation of tangent line if slope is m and passes (a,f(a))?
y−f(a)=m(x−a).
Define inflection point in terms of f’‘(x).
Where concavity changes sign (f’’ changes from + to − or vice versa).
If f’(x)=0 for all x, what does that say about f(x)?
f is constant.
Mean Value Theorem condition in short?
If f is continuous on [a,b] and differentiable on (a,b), there exists c∈(a,b) with f’(c)= [f(b)−f(a)] / [b−a].
Given f’(x)=3x², find original function f(x) ignoring constants.
f(x)=x^3 (plus a constant).
If velocity is derivative of position, v(t)=s’(t), then acceleration is?
a(t)=v’(t)=s’‘(t).
Max or min is found by checking critical points and endpoints if domain is closed. T/F?
True.
For function y=x², derivative is 2x. Where is slope=0?
At x=0 (lowest point).
At x=2, if f’(2)=0 and f’‘(2)>0, then x=2 is local?
Minimum.
At x=−1, if f’(−1)=0 and f’’(−1)=−3, local?
Maximum.
If f’(x)>0 for all x in (a,b), is f increasing or decreasing?
Increasing.
If f’(x)<0 for all x in (a,b), is f increasing or decreasing?
Decreasing.
If f’‘(x)>0, the graph of f is concave up or down?
Concave up.
If f’‘(x)<0, the graph is concave up or down?
Concave down.
Define indefinite integral ∫ f(x) dx.
All antiderivatives of f(x) + a constant C.
Define definite integral ∫(a to b) f(x) dx.
Area under f(x) from x=a to x=b (if f≥0).
Fundamental Theorem of Calculus, Part 1 in short form?
d/dx ∫(a to x) f(t) dt = f(x).
Fundamental Theorem of Calculus, Part 2 formula?
∫(a to b) f(x) dx=F(b)−F(a) where F’(x)=f(x).
Antiderivative of f(x)=xⁿ?
x^(n+1)/(n+1) + C, n≠−1.
Antiderivative of f(x)=1/x?
ln|x| + C.
Antiderivative of sin x?
−cos x + C.
Antiderivative of cos x?
sin x + C.
Antiderivative of sec² x?
tan x + C.
Antiderivative of e^x?
e^x + C.
Evaluate ∫ dx/x.
ln|x| + C.
Evaluate ∫ e^x dx.
e^x + C.
Evaluate ∫ sin x dx.
−cos x + C.
Evaluate ∫ cos x dx.
sin x + C.
Evaluate ∫ (3x²) dx.
x³ + C.
For definite integral ∫(0 to 2) x dx, result?
[x²/2]_0^2 = (4/2)−0=2.
Evaluate ∫(1 to 2) 1/x dx.
[ln|x|]_1^2=ln2−ln1=ln2.
Evaluate ∫(0 to π/2) cos x dx.
[sin x]_0^(π/2)=1−0=1.
If F’(x)=f(x), then ∫(a to b) f(x) dx=?
F(b)−F(a).
What is the interpretation of ∫(a to b) f(x) dx if f≥0?
Area under the curve from x=a to x=b.
Integration by parts formula?
∫ u dv = uv − ∫ v du.
u-substitution formula in short?
If u=g(x), then ∫ f(g(x))g’(x) dx=∫ f(u) du.
Evaluate ∫ 2x e^(x²) dx using substitution.
Let u=x², du=2x dx → result=∫ e^u du = e^u + C=e^(x²)+C.
Evaluate ∫ x cos(x²) dx.
Use u=x² => du=2x dx => (1/2)∫ cos(u) du => (1/2) sin(u)+C => (1/2) sin(x²)+C.
Integration by parts example: ∫ x e^x dx. Let u=x, dv=e^x dx.
uv−∫v du = x e^x − ∫ e^x(1) dx = x e^x − e^x + C= e^x(x−1)+C.
Evaluate ∫ sec² x dx.
tan x + C.
Evaluate ∫(0 to π/4) tan x dx.
(1/2)ln(2).
Evaluate ∫(1 to e) ln x dx by integration by parts.
1.
Evaluate derivative of F(x)=∫(2 to x) t² dt using Fundamental Theorem.
F’(x)=x².
If f(x)=d/dx[∫(0 to x) e^(t^2) dt], then f(x)=?
e^(x^2).
Improper integral example: ∫(1 to ∞) 1/x² dx=?
1.
Define indefinite integral vs. definite integral in short.
Indefinite: general antiderivative plus C, definite: real number from a→b.
What is a Riemann sum concept for ∫(a to b) f(x) dx?
Limit of sum.
Evaluate ∫(1 to ∞) 1/x² dx=?
Limit b→∞ ∫(1 to b) 1/x² dx= limit b→∞ [-1/x]_1^b= [−1/b +1/1]=1.
What is a Riemann sum concept for ∫(a to b) f(x) dx?
Limit of sum f(xᵢ*)Δx as Δx→0, partitioning [a,b].
Arc length formula (basic version) for y=f(x) from x=a to x=b?
∫(a to b) √[1+(f’(x))²] dx.
Volume of revolution using disk method example formula?
V=π∫(a to b) [R(x)]² dx, where R(x) is radius.
Derivative of parametric function y(t)= e^t, x(t)= t => dy/dx=?
(dy/dt)/(dx/dt)= (e^t)/(1)= e^t.
Define average value of f(x) on [a,b].
(1/(b−a)) ∫(a to b) f(x) dx.
Evaluate ∫(0 to π/2) sin x dx quickly.
[−cos x]_0^(π/2)= (−cos(π/2))−(−cos(0))= (−0)−(−1)=1.
Evaluate ∫(0 to 2) 3 dx.
3x from 0 to 2=6.
If dy/dx=3x², then y=? (Ignoring constant).
x³.
Limit definition of derivative at x=a
f’(a)=lim(h→0) [f(a+h)−f(a)] / h
Limit definition of continuity at x=a
f is continuous at a if lim(x→a) f(x)=f(a)
Epsilon-delta definition of limit (in words)
For every ε>0, there is δ>0 so |x−a|<δ ⇒ |f(x)−L|<ε
L’Hôpital’s Rule condition (0/0 or ∞/∞)
If lim gives 0/0 or ∞/∞, then lim(f/g)=lim(f’/g’) if that latter limit exists
Power rule for derivatives
d/dx [x^n] = n x^(n−1)
Sum rule for derivatives
d/dx [f(x)+g(x)] = f’(x)+g’(x)
Product rule for derivatives
d/dx [u·v] = u’v + uv’
Quotient rule for derivatives
d/dx [u/v] = [u’v − uv’] / v²
Chain rule for derivatives
d/dx [f(g(x))] = f’(g(x))·g’(x)
Derivative of e^x
(e^x)’= e^x
Derivative of a^x (a>0)
(a^x)’ = a^x ln a
Derivative of ln(x), x>0
(ln x)’=1/x
Derivative of sin x
(sin x)’=cos x
Derivative of cos x
(cos x)’=−sin x
Derivative of tan x
(tan x)’=sec² x
Derivative of sec x
(sec x)’=sec x tan x
Derivative of arcsin x
(arcsin x)’=1/√(1−x²)
Derivative of arctan x
(arctan x)’=1/(1+x²)
Mean Value Theorem short statement
If f is continuous on [a,b], differentiable on (a,b), ∃ c with f’(c)= [f(b)−f(a)]/(b−a)
Definition of critical point
Where f’(x)=0 or undefined, and f(x) is defined
Second derivative test for local min
If f’(a)=0 and f’‘(a)>0 ⇒ local minimum at x=a
Second derivative test for local max
If f’(a)=0 and f’‘(a)<0 ⇒ local maximum at x=a
Concavity from the second derivative
f’‘(x)>0 ⇒ concave up, f’‘(x)<0 ⇒ concave down
Definition of inflection point
Where the function changes concavity (f’’ changes sign)
Definition of indefinite integral
∫ f(x) dx = F(x) + C, where F’(x)=f(x)
Power rule for integrals
∫ x^n dx = x^(n+1)/(n+1) + C, n≠−1
Integral of e^x
∫ e^x dx= e^x + C
Integral of 1/x, x≠0
∫ (1/x) dx= ln|x| + C
Integral of sin x
∫ sin x dx= −cos x + C
Integral of cos x
∫ cos x dx= sin x + C
Integral of sec² x
∫ sec² x dx= tan x + C
Integral of 1/(1+x²)
∫ 1/(1+x²) dx= arctan x + C
Basic u-substitution formula
If u=g(x), du=g’(x)dx ⇒ ∫ f(g(x))g’(x) dx= ∫ f(u) du
Integration by parts formula
∫ u dv= uv − ∫ v du
Fundamental Theorem of Calculus, Part 1
d/dx [∫(a to x) f(t) dt]= f(x)
Fundamental Theorem of Calculus, Part 2
∫(a to b) f(x) dx= F(b)−F(a), where F’(x)=f(x)
Definition of definite integral via Riemann sum
∫(a to b) f(x) dx= lim(n→∞) Σ f(xᵢ*)Δx
Average value of a function f on [a,b]
Favg= (1/(b−a)) ∫(a to b) f(x) dx
d/dx [∫(c to g(x)) f(t) dt] by chain rule
f(g(x))·g’(x)
L’Hôpital’s Rule formula for 0/0 or ∞/∞
lim(x→a) f(x)/g(x)= lim(x→a) f’(x)/g’(x) if the latter limit exists
Derivative of ln|x|
1/x for x≠0
Integral of sec x tan x
∫ sec x tan x dx= sec x + C
Integral of sec x
∫ sec x dx= ln|sec x + tan x| + C
Integral of csc² x
∫ csc² x dx= −cot x + C
Integral of tan x
∫ tan x dx= −ln|cos x| + C
Arc length formula for y=f(x), x in [a,b]
∫(a to b) √[1 + (f’(x))²] dx
Volume by disk method if rotating y=f(x)≥0 about x-axis
V= π ∫(a to b) [f(x)]² dx
Volume by shell method for rotating around y-axis
V= 2π ∫(a to b) x·f(x) dx
d/dx [arcsin(x)]?
1/√(1−x²)
d/dx [arccos(x)]?
−1/√(1−x²)
d/dx [arctan(x)]?
1/(1+x²)
d/dx [arcsec(x)] (|x|>1)?
1/[|x|√(x²−1)] with sign considerations
Integration: ∫ dx/(a² + x²)
(1/a) arctan(x/a) + C
Integration: ∫ dx/√(a²−x²)
arcsin(x/a) + C
Integration: ∫ dx/√(x²±a²)
Use ln|x+√(x²±a²)| forms or trig/hyperbolic sub
Definition of partial fractions approach
Decompose rational expression into simpler fraction terms
Integral of 1/(1 + x²) from 0 to 1
[arctan x]_0^1= π/4.
d/dx [csc x]?
−csc x cot x
d/dx [cot x]?
−csc² x
Integral of csc x cot x dx
−csc x + C
Define Newton’s Method iteration formula in short
x_{n+1}= x_n − f(x_n)/f’(x_n)
Linear approximation formula at x=a
f(x)≈f(a)+f’(a)(x−a)
Differential dy if y=f(x)
dy=f’(x) dx
Integration by parts in short: ∫u dv
uv−∫v du
Second Fundamental Theorem version: d/dx [∫(g(x) to h(x)) f(t) dt]
f(h(x))h’(x)−f(g(x))g’(x)
If f’(x)=0 for all x in an interval, then f is?
Constant in that interval.
If f’‘(x)=0 for all x in an interval, then f’(x) is?
Constant.
Relation for indefinite integrals: ∫[f’(x)/f(x)] dx
ln|f(x)| + C
Bernoulli’s differential equation form (basic mention)
dy/dx + P(x)y=Q(x)y^n
dy/dx= k·y means y=? (Solving simple differential eq)
y=Ce^(kx)
Sine integral approximation near 0: sin x≈x. So (sin x)/x→1 as x→0. T/F?
True.
Definition: slope of tangent line at x=a is?
f’(a).
Definition: slope of normal line at x=a is?
−1/f’(a) (perpendicular slope).
If f’(x)>0 on (a,b), then f is (inc/dec)?
Increasing.
If f’(x)<0 on (a,b), f is?
Decreasing.
If f’‘(x)>0 on an interval, f is concave (up/down)?
Up.
If f’‘(x)<0, concavity is?
Down.
Derivative of parametric eq: y’(t)/x’(t). T/F?
True. dy/dx= (dy/dt)/(dx/dt).
Arc length parametric formula: L=?
∫√[(dx/dt)² + (dy/dt)²] dt over t-range.
Simple example of partial fraction: ∫ 1/[(x−1)(x+1)] dx =>?
½ ln| (x−1)/(x+1) | + C.
Area under f(x)≥0 from x=a to b is ∫(a to b) f(x) dx. T/F?
True.
d/dx [ln|cos x|] =?
(1/cos x)(−sin x)= −tan x.
d/dx [sin−1(2x)] by chain rule =>?
1/√(1−(2x)²) · 2= 2/√(1−4x²).
Derivative of f(x)= ln(x² +1) =>?
(1/(x²+1))·2x= 2x/(x²+1).
Define indefinite integral of sec²(ax) dx =>?
(1/a) tan(ax)+C.
Define indefinite integral of 1/(ax+b) dx =>?
(1/a) ln|ax+b|+C.
Define indefinite integral of cos(kx) dx =>?
(1/k) sin(kx)+C.
Define indefinite integral of sin(kx) dx =>?
−(1/k) cos(kx)+C.
Definition of logistic differential eq: dy/dt= ky(1−y/M)? T/F?
True, a standard logistic form.
Integration technique: ∫ 2x (x²+1)^5 dx => let u=(x²+1). T/F?
True, du=2x dx => integral= ∫ u^5 du= u^6/6 +C.
Define indefinite integral: ∫ sec x tan x dx =>?
sec x + C.
Define indefinite integral: ∫ csc x cot x dx =>?
−csc x + C.
List the 4 steps in basic u-substitution
Identify inside function, let u=…, compute du, rewrite integral, integrate, back-substitute.
Define derivative of arcsin(g(x)): chain rule =>?
(g’(x))/√(1−(g(x))²).
Define derivative of arccos(g(x)) =>?
−g’(x)/√(1−(g(x))²).
Define derivative of arctan(g(x)) =>?
g’(x)/(1+(g(x))²).
For x>0, derivative of ln(g(x)) =>?
(g’(x))/(g(x)).
Integration by parts: ∫ x e^x dx =>?
x e^x − ∫ e^x dx= x e^x − e^x + C= e^x(x−1)+C.
Definition: ∫(a to b) f’(x) dx =>?
f(b)−f(a).
Concise formula: derivative of f(g(h(x))) =>?
f’(g(h(x)))·g’(h(x))·h’(x).
Find the derivative of f(x)=x²+3x.
2x + 3.
Evaluate the limit: lim(x→∞) (2x)/(x+1).
2.
Find d/dx of f(x)=√x = x^(1/2).
(1/2)x^(-1/2).
Compute ∫ 3x² dx.
x³ + C.
Evaluate lim(x→0) (sin x)/x.
1.
Find the derivative of f(x)=sin(2x).
2 cos(2x).
Compute ∫ (1/x) dx (x≠0).
ln|x| + C.
Evaluate lim(x→∞) 1/x².
0.
Find the derivative of f(x)=e^x + 5.
e^x.
Compute ∫ cos x dx.
sin x + C.
What is d/dx of ln(x) for x>0?
1/x.
Evaluate ∫ (4x) dx.
2x² + C.
Find derivative: f(x)=5/x.
f’(x)=−5/x².
Evaluate lim(x→0) (1−cos x)/x² (a known limit).
1/2.
Find the slope of f(x)=x³ at x=2.
Derivative is 3x², so slope=3(2)²=12.
Compute ∫ sec² x dx.
tan x + C.
Evaluate lim(x→∞) (3x²)/(6x²+1).
Divide top/bottom by x² => 3/6=1/2.
Find derivative of f(x)=3x−7.
3.
Compute ∫ (2x+1) dx.
x² + x + C.
Evaluate lim(x→3) (x²−9)/(x−3) by factorization.
x²−9=(x−3)(x+3) => limit=3+3=6.
Find derivative of f(x)=sin x + e^x.
cos x + e^x.
Compute ∫ (1 + 2x) dx.
x + x² + C.
Evaluate lim(x→0) x/(sin x).
1.
Find derivative: g(x)=ln(5x).
1/(5x)*5=1/x.
Compute ∫ dx/(x+1).
ln|x+1| + C.
Evaluate lim(x→∞) (x³)/(2x³ +1).
1/2.
Find derivative: f(x)=x² e^x (product rule).
2x e^x + x² e^x = e^x(2x + x²).
Compute ∫ 4 dx.
4x + C.
Evaluate lim(h→0) [ (2+h)³ − 2³ ] / h.
Use expand: (8+12h+6h²+h³−8)/h => (12h+6h²+h³)/h =>12+6h+h² => as h→0 => 12.
Find derivative f(x)=x^(1/3). (Use power rule, n=1/3.)
(1/3)x^(-2/3).
Compute ∫ e^x dx.
e^x + C.
Evaluate lim(x→π/2) cos x.
0.
Find derivative f(x)=3 cos(2x).
3(−sin(2x))·2=−6 sin(2x).
Compute ∫ 1/(x²+1) dx.
arctan x + C.
Evaluate lim(x→0) (ln(1 + x))/x.
1 (by expansion or L’Hôpital).
Find derivative: f(x)=x^(4).
4x³.
Compute ∫ cos(2x) dx using substitution.
(1/2) sin(2x) + C.
Evaluate lim(x→∞) e^(-x).
0.
Find derivative f(x)=3/(x+2).
Rewrite 3(x+2)^(-1) => derivative=3(−1)(x+2)^(-2)(1)=−3/(x+2)².
Compute ∫ x dx.
x²/2 + C.
Evaluate lim(x→1) (x²−1)/(x−1).
Factor numerator => (x−1)(x+1)/(x−1)=x+1 => at x=1 =>2.
Find derivative: f(x)=ln(x²+1).
(1/(x²+1))·2x= 2x/(x²+1).
Compute ∫ 6x dx.
3x² + C.
Evaluate lim(x→∞) (5x−2)/(−2x+1).
Divide top/bottom by x => (5−2/x)/ (−2+1/x)=> ratio= −(5/2)? Wait carefully: top is 5, bottom is −2 => limit= (5)/ (−2)= −5/2.
Find derivative: f(x)=tan x.
sec² x.
Evaluate lim(x→0) (1− e^(-x))/x.
By expansion or L’Hôpital => (1−(1−x+…))/x => or L’Hôpital => derivative top e^(-x)(−1), bottom=1 => at 0 => 1 => Actually check sign. Let’s do L’Hop: top derivative= e^(-x), but there’s negative sign? derivative(1−e^(-x))= e^(-x). Evaluate at x=0 => e^0=1. So limit=1.
Find derivative: f(x)=x² sin x (product rule).
2x sin x + x² cos x.
Compute ∫(1 to 2) (2x) dx.
[x²]_1^2= (4)−1=3.
Evaluate derivative at x=2 if f(x)= (x+1)³ => f’(x)= 3(x+1)² => f’(2)=?
3(2+1)²=3(3)²=3·9=27.
