RepetoIf \(f\) is differentiable at \(a\), what is the equation of its tangent line there?
\(y-f(a)=f'(a)(x-a)\).
Calculus I: derivatives and integrals
320 questions in English on limits, derivatives, applications and elementary integrals.
- 320 cards
- By Repeto
- Free
What the cards look like
If \(f\) is continuous on \([a,b]\) and \(F'=f\), how is \(\int_a^b f(x)\,dx\) evaluated?
\(F(b)-F(a)\).
Revolve the region \(0\le x\le4\), \(0\le y\le\sqrt{x}\), around the x-axis. What is the volume?
\(8\pi\) cubic units.
On an interval excluding \(0\), what is \(\int\frac{1}{x}\,dx\)?
\(\ln\lvert x\rvert+C\).
With \(x\) in radians, what is \(\frac{d}{dx}\cos x\)?
\(-\sin x\).
Evaluate \(\lim_{x\to\infty}(1+2/x)^x\).
\(e^2\).
All 320 cards
Every card in the deck, in study order.
Question Answer
If \(f\) is differentiable at \(a\), what is the equation of its tangent line there? \(y-f(a)=f'(a)(x-a)\).
If \(f\) is continuous on \([a,b]\) and \(F'=f\), how is \(\int_a^b f(x)\,dx\) evaluated? \(F(b)-F(a)\).
Revolve the region \(0\le x\le4\), \(0\le y\le\sqrt{x}\), around the x-axis. What is the volume? \(8\pi\) cubic units.
On an interval excluding \(0\), what is \(\int\frac{1}{x}\,dx\)? \(\ln\lvert x\rvert+C\).
With \(x\) in radians, what is \(\frac{d}{dx}\cos x\)? \(-\sin x\).
Evaluate \(\lim_{x\to\infty}(1+2/x)^x\). \(e^2\).
Find \(\int e^{4x}\,dx\). \(\frac14e^{4x}+C\).
Differentiate \(f(x)=(x^2+2)^{-3}\) for real \(x\). \(f'(x)=-6x(x^2+2)^{-4}\).
Evaluate \(\int_0^{\ln 4}e^x\,dx\). \(3\).
Differentiate \(f(x)=x e^{3x}\) for real \(x\). \(f'(x)=e^{3x}(1+3x)\).
Differentiate \(f(x)=\ln x\) on \(x>0\). \(f'(x)=1/x\).
Does \(\lim_{x\to a}f(x)=+\infty\) assert a finite real limit? No. It describes unbounded positive growth near \(a\).
For continuity on \([a,b]\), which one-sided limit must equal \(f(b)\) at the right endpoint? The left-hand limit \(\lim_{x\to b^-}f(x)\).
Evaluate \(\lim_{x\to\infty}\frac{\ln x}{x}\). \(0\).
What identity defines an even function on a domain symmetric about zero? \(f(-x)=f(x)\) for every input in its domain.
With \(x\) in radians and \(\cos x\ne0\), what is \(\frac{d}{dx}\tan x\)? \(\sec^2 x\).
If \(f\) is continuously differentiable near \(a\) and \(f'(a)\ne0\), what is the derivative of its local inverse at \(f(a)\)? \((f^{-1})'(f(a))=1/f'(a)\).
If \(F'\) is continuous on \([a,b]\), what does \(\int_a^b F'(x)\,dx\) measure? \(F(b)-F(a)\), the net change in \(F\).
Why must \(x^3+x-5=0\) have a solution in \((1,2)\)? The polynomial is continuous, and its values at \(1\) and \(2\) are \(-3\) and \(5\), so the IVT guarantees a zero between them.
Differentiate \(f(x)=\sin(4x^2)\), using radians. \(f'(x)=8x\cos(4x^2)\).
What does it mean for \(f(c)\) to be an absolute maximum of \(f\) on a domain \(D\)? \(c\in D\) and \(f(c)\ge f(x)\) for every \(x\in D\).
Suppose \(f\) has a differentiable local inverse, \(f(2)=7\), and \(f'(2)=5\). What is \((f^{-1})'(7)\)? \(1/5\).
For a positive differentiable function \(y=y(x)\), what is \(\frac{d}{dx}\ln y\)? \(y'/y\).
If \(f\) is continuous and \(g\) is differentiable, what is \(\frac{d}{dx}\int_a^{g(x)}f(t)\,dt\) where the integral is defined? \(f(g(x))g'(x)\).
Revolve the region under continuous \(h(x)\ge0\) on \(0\le a\le x\le b\) around the y-axis. What is its volume by shells? \(V=2\pi\int_a^b xh(x)\,dx\).
What is the area enclosed between \(y=x\) and \(y=x^2\) for \(0\le x\le1\)? \(1/6\) square units.
Find \(\frac{d}{dx}\int_1^{x^2}e^t\,dt\). \(2xe^{x^2}\).
What is the tangent slope to \(x^2+y^2=169\) at \((5,12)\)? \(-5/12\).
In radians, find \(\int\cos(5x)\,dx\). \(\frac15\sin(5x)+C\).
Does a quotient with numerator tending to \(2\) and denominator tending to \(0\) have a \(0/0\) indeterminate form? No. One must inspect the denominator's sign and one-sided behavior; L'Hôpital's \(0/0\) rule does not apply directly.
For \(f(x)=2x^2-5x\), what is \(f'(3)\)? \(7\). The difference quotient at \(3\) simplifies to \(7+2h\).
On \(x^2+xy+y^2=7\), where \(x+2y\ne0\), what is \(dy/dx\)? \(-\frac{2x+y}{x+2y}\).
Under \(u=g(x)\), which bounds replace \(x=a\) and \(x=b\) in a definite integral? The lower bound becomes \(g(a)\) and the upper bound becomes \(g(b)\), preserving their order.
Evaluate \(\int_0^2 2x(x^2+1)^2\,dx\). \(124/3\). With \(u=x^2+1\), the bounds are \(1\) and \(5\).
Differentiate \(f(x)=x\sin x\), with \(x\) in radians. \(f'(x)=\sin x+x\cos x\).
Where is the inflection point of \(f(x)=x^3-12x\)? \((0,0)\), since \(f''(x)=6x\) changes from negative to positive at \(0\).
Can a strictly increasing differentiable function have derivative zero at a point? Yes. \(f(x)=x^3\) is strictly increasing and has \(f'(0)=0\).
Evaluate \(\lim_{x\to1}(2x+5)/(x^2+3)\). \(7/4\).
In radians, evaluate \(\int_0^\pi\sin x\,dx\). \(2\).
Find \(\frac{d}{dx}\int_x^5(t^2+1)\,dt\). \(-(x^2+1)\).
Differentiate \(f(x)=\cos(5x-2)\), using radians. \(f'(x)=-5\sin(5x-2)\).
Is \((uv)'=u'v'\) generally valid for differentiable \(u,v\)? No. The product rule is \(u'v+uv'\).
With \(x\) in radians and \(\sin x\ne0\), what is \(\frac{d}{dx}\cot x\)? \(-\csc^2 x\).
Evaluate \(\lim_{x\to-2}1/(x+2)^2\). \(+\infty\) from both sides.
A vertical cylinder of fixed radius \(2\) cm fills at \(8\pi\) cubic centimeters per second. With \(V=\pi r^2h\), how fast does the water height rise? \(2\) centimeters per second.
Revolve the region \(0\le x\le1\), \(0\le y\le1-x\), around the y-axis. What is the volume? \(\pi/3\) cubic units.
If \(c\) is constant and \(f\) is differentiable, what is \((cf)'\)? \(cf'\).
With radian input, evaluate \(\lim_{x\to0}\sin x/x\). \(1\).
A rectangle has perimeter \(28\) m and one side \(x\) m. What is its area \(A(x)\) for \(0<x<14\)? \(A(x)=x(14-x)\) square meters.
How is \(f'(a)\) defined by a difference quotient at an interior point? \(f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\), provided the limit exists as a finite real number.
If \(f\) and \(g\) are differentiable, what is \((f+g)'\)? \(f'+g'\).
What is the mesh of a partition of \([a,b]\)? The width of its widest subinterval.
For \(x>0\), find \(\int\frac{\ln x}{x}\,dx\). \(\frac12(\ln x)^2+C\).
Evaluate \(\lim_{x\to-2}(3x^2-x+4)\). \(18\).
For \(n\) equal pieces of \([a,b]\), what is the right endpoint \(x_i\) of piece \(i\)? \(x_i=a+i\Delta x\), where \(\Delta x=(b-a)/n\) and \(i=1,\ldots,n\).
At a differentiable point, what derivative value corresponds to a horizontal tangent? \(f'(a)=0\).
Differentiate \(f(x)=x^{5/3}\) on \(x>0\). \(f'(x)=\frac53x^{2/3}\).
When does a finite two-sided limit at \(a\) exist in terms of the one-sided limits? Both one-sided limits must exist and equal the same finite number.
Find \(\frac{d}{dx}\int_x^{2x}t^2\,dt\). \(7x^2\).
If \(v'(t)=4t+1\) and \(v(0)=-2\), what is \(v(t)\)? \(v(t)=2t^2+t-2\).
Show the other 260 cards
When \(f'(x_n)\ne0\), what is the Newton iteration for solving \(f(x)=0\)? \(x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\).
A position function is \(s(t)=t^2+2t\) meters. What is the average velocity from \(t=1\) to \(t=4\) seconds? \(7\) meters per second.
On \(x^2+y^2=169\), where \(y\ne0\), what is \(dy/dx\)? \(-x/y\).
For continuity on \([a,b]\), which one-sided limit must equal \(f(a)\) at the left endpoint? The right-hand limit \(\lim_{x\to a^+}f(x)\).
What convergence requirement defines the Riemann integral of a bounded function on \([a,b]\)? The Riemann sums must tend to one finite value as the partition mesh tends to zero, regardless of the allowed sample points and partitions.
Revolve the region \(0\le x\le2\), \(x\le y\le2\), around the x-axis. What is the volume? \(16\pi/3\) cubic units.
If \(g(1)=4\), \(g'(1)=-2\), and \(f'(4)=3\), what is \((f\circ g)'(1)\)? \(-6\).
If \(F'=f\), \(f\) is continuous, and \(g'\) is continuous, what is \(\int f(g(x))g'(x)\,dx\) on a suitable interval? \(F(g(x))+C\).
If \(g\) is differentiable at \(x\) and \(f\) at \(g(x)\), what is \(\frac{d}{dx}f(g(x))\)? \(f'(g(x))g'(x)\).
Differentiate \(f(x)=e^{x^2+3}\) for real \(x\). \(f'(x)=2xe^{x^2+3}\).
For continuous velocity \(v(t)\) on \([a,b]\), \(a<b\), what integral gives total distance traveled? \(\int_a^b\lvert v(t)\rvert\,dt\).
With \(x\) in radians, what is \(\frac{d}{dx}\sin x\)? \(\cos x\).
Which regularity and interval conditions are required for the Intermediate Value Theorem on \([a,b]\), with \(a<b\)? The function must be continuous throughout the closed interval \([a,b]\).
For \(f(x)=2\cos x+3\sin x\), with radian input, what is \(f'(0)\)? \(3\).
Differentiate \(f(x)=x^{-4}\) for \(x\ne0\). \(f'(x)=-4x^{-5}\).
For real \(p\ne-1\) and \(x>0\), what is \(\int x^p\,dx\)? \(\frac{x^{p+1}}{p+1}+C\).
For positive \(u(x)\), what transformation helps evaluate an indeterminate power \(u(x)^{v(x)}\)? Set \(y=u(x)^{v(x)}\), evaluate the limit of \(\ln y=v(x)\ln u(x)\), then exponentiate when that limit is finite.
Can a function with a jump discontinuity at \(a\) be differentiable at \(a\)? No, because differentiability would imply continuity.
Does Newton's method converge to a root from every starting point for every differentiable function? No. It can fail, cycle, diverge, or encounter a zero derivative.
If \(f\) is differentiable at \(a\), must it be continuous there? Yes.
For \(f\) differentiable at \(a\), what is its linearization \(L(x)\) at \(a\)? \(L(x)=f(a)+f'(a)(x-a)\).
Differentiate \(f(x)=x\ln x\) for \(x>0\). \(f'(x)=\ln x+1\).
In radians, find \(\int\sin x\,dx\). \(-\cos x+C\).
Use four equal subintervals to find the left-endpoint sum for \(f(x)=x^2\) on \([0,2]\). \(7/4\), an approximation to the integral.
Differentiate \(f(x)=\ln\lvert 2x-1\rvert\) for \(x\ne1/2\). \(f'(x)=2/(2x-1)\).
What is the real domain of \(f(x)=1/(x^2-9)\)? All real \(x\) except \(-3\) and \(3\).
What identity defines an odd function on a domain symmetric about zero? \(f(-x)=-f(x)\) for every input in its domain.
Differentiate \(f(x)=\sin(e^{2x})\), using radians. \(f'(x)=2e^{2x}\cos(e^{2x})\).
Evaluate \(\lim_{x\to4^+}1/(x-4)\). \(+\infty\); the values grow without bound.
If \(f\) is continuous near \(c\) and \(f'>0\) on both sides of \(c\), does a possible zero derivative at \(c\) make it a local extremum? No. The function increases through \(c\).
What is the absolute minimum of \(f(x)=x^2-6x+11\) on \([0,5]\)? \(2\), attained at \(x=3\).
Which property fails for \(f(x)=\lvert x\rvert\) at \(0\), despite continuity there? Differentiability: the left and right derivatives are \(-1\) and \(1\).
If \(\lim_{x\to a}f(x)=L\), must \(f(a)=L\)? No. The limit concerns values near \(a\); \(f(a)\) may differ or be undefined.
If \(f\) is continuous on \([a,b]\) and \(N\) lies strictly between \(f(a)\) and \(f(b)\), what does the IVT guarantee? Some \(c\in(a,b)\) satisfies \(f(c)=N\).
What is the domain of a real-valued function? The set of allowed input values.
Is \(dy=f'(x)\,dx\) always equal to \(f(x+dx)-f(x)\)? No. It is a linear approximation to that change; nonlinear terms generally remain.
On an interval where \(\cos x\ne0\), find \(\int\sec x\tan x\,dx\). \(\sec x+C\).
Differentiate \(f(x)=x^x\) for \(x>0\). \(f'(x)=x^x(\ln x+1)\).
After converting a definite integral completely to \(u\), should its endpoints still be the original x-values? No. Use u-values as bounds, or return the antiderivative to \(x\) before evaluating at the original x-values.
Under Rolle's theorem's hypotheses, what interior point is guaranteed? At least one \(c\in(a,b)\) with \(f'(c)=0\).
For a real constant \(c\), what is \(\frac{d}{dx}c\)? \(0\).
Revolving \(0\le y\le R(x)\), with continuous \(R\ge0\) on \([a,b]\), around the x-axis gives what volume? \(V=\pi\int_a^b R(x)^2\,dx\).
What is the linearization of \(f(x)=\sqrt{x}\) at \(a=25\)? \(L(x)=5+(x-25)/10\).
If \(f\) is continuous, what is \(\frac{d}{dx}\int_x^b f(t)\,dt\) for interior \(x\)? \(-f(x)\).
For differentiable functions \(u\) and \(v\), what is \((uv)'\)? \(u'v+uv'\).
Is \(x=0\) a critical number of \(f(x)=1/x\)? No. It is outside the domain of \(f\).
If \(f''<0\) throughout an interval, what is the graph's concavity there? Concave down; the slopes \(f'\) decrease.
For continuous \(f\) on \([a,b]\), \(a<b\), what integral gives the total geometric area between its graph and the x-axis? \(\int_a^b\lvert f(x)\rvert\,dx\).
Evaluate \(\lim_{x\to9}(\sqrt{x}-3)/(x-9)\). \(1/6\).
If \(F'(x)=6x^2-4x\) and \(F(1)=5\), what is \(F(x)\)? \(F(x)=2x^3-2x^2+5\).
If an object's velocity is \(-8\) meters per second, what is its speed? \(8\) meters per second.
Using the tangent at \(0\), what linear approximation estimates \(e^{0.03}\)? \(e^{0.03}\approx1.03\).
Two objects move on perpendicular axes with distances \(x=6\), \(y=8\) meters from the origin and rates \(x'=2\), \(y'=1\) m/s. What is their separation rate? \(2\) m/s, from \(s'=\frac{xx'+yy'}{\sqrt{x^2+y^2}}\).
What conditions on \(f\) allow the Mean Value Theorem on \([a,b]\), where \(a<b\)? Continuity on \([a,b]\) and differentiability on \((a,b)\).
For differentiable \(u\) and \(v\), with \(v(x)\ne0\), what is \((u/v)'\)? \(\frac{u'v-uv'}{v^2}\).
If \(T'(5)=-2\) degrees Celsius per minute, what is the instantaneous temperature trend at \(t=5\)? Temperature is decreasing at \(2\) degrees Celsius per minute at that instant.
What hypotheses guarantee that a real function attains both an absolute maximum and an absolute minimum on \([a,b]\)? Continuity on the closed, bounded interval \([a,b]\).
Does the Intermediate Value Theorem by itself guarantee exactly one point with a specified intermediate value? No. It guarantees existence, not uniqueness.
Why must an optimization problem specify a feasible domain? It records which variable values satisfy the constraints and physical restrictions.
Revolve the region \(0\le x\le2\), \(0\le y\le x\), around the x-axis. What is the volume? \(8\pi/3\) cubic units.
Does \(f(x)=x\) attain an absolute maximum on \((0,1)\)? No. Its supremum is \(1\), but \(1\) is not attained on that domain.
For \(x>0\), find \(\int\sqrt{x}\,dx\). \(\frac{2}{3}x^{3/2}+C\).
If \(u(2)=3\), \(u'(2)=4\), \(v(2)=-1\), and \(v'(2)=5\), what is \((uv)'(2)\)? \(11\).
Find \(\int e^x\,dx\). \(e^x+C\).
For a positive integer \(n\), what is \(\frac{d}{dx}x^n\)? \(nx^{n-1}\), for all real \(x\).
For continuous radii \(0\le r(x)\le R(x)\), what volume is formed by washers perpendicular to the x-axis on \([a,b]\)? \(V=\pi\int_a^b(R(x)^2-r(x)^2)\,dx\).
A particle has velocity \(v(t)=2t-4\) m/s for \(0\le t\le5\) s. What total distance does it travel? \(13\) m; split the absolute-value integral at \(t=2\).
For \(x\ne0\), differentiate \(f(x)=(x^3+2x)/x\). \(f'(x)=2x\) on the original domain \(x\ne0\).
If \(f\) is continuous on \([a,b]\) and \(A(x)=\int_a^x f(t)\,dt\), what is \(A'(x)\) for \(a<x<b\)? \(A'(x)=f(x)\).
If a solid has continuous cross-sectional area \(A(x)\) perpendicular to the x-axis for \(a\le x\le b\), what is its volume? \(V=\int_a^b A(x)\,dx\).
When \([a,b]\), with \(a<b\), is divided into \(n\) equal pieces, what is the width \(\Delta x\)? \(\Delta x=(b-a)/n\).
Differentiate \(f(x)=\sqrt{2x+5}\) for \(x>-5/2\). \(f'(x)=1/\sqrt{2x+5}\).
Differentiate \(f(x)=e^x\) for real \(x\). \(f'(x)=e^x\).
Differentiate \(f(x)=\sqrt{x}\) for \(x>0\). \(f'(x)=1/(2\sqrt{x})\).
If \(f'=g'\) throughout an open interval \(I\), how are \(f\) and \(g\) related on \(I\)? They differ by one constant: \(f(x)=g(x)+C\) for all \(x\in I\).
For an increasing continuous function and equal subintervals, how do left- and right-endpoint sums compare with its integral? The left sum is a lower bound and the right sum is an upper bound.
In radians, find \(\int\sin(3x)\,dx\). \(-\frac13\cos(3x)+C\).
If \(f'(x)<0\) throughout an interval, how does \(f\) vary there? It is strictly decreasing.
Evaluate \(\int_0^3\frac{1}{9+x^2}\,dx\). \(\pi/12\).
What does \(f''(x)\) mean? The derivative of the first derivative, wherever that derivative exists.
If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a)\), what follows about \(f\circ g\)? It is continuous at \(a\).
What is the real domain of \(f(x)=\sqrt{7-2x}\)? \(x\le 7/2\).
If \(f\) is continuous on \([a,b]\), \(a<b\), what does the integral mean value theorem guarantee? At least one \(c\in[a,b]\) satisfies \(f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx\).
Differentiate \(f(x)=\sin^2 x\), using radians. \(f'(x)=2\sin x\cos x\).
What is the tangent line to \(y=x^3\) at \(x=2\)? \(y-8=12(x-2)\), or \(y=12x-16\).
Differentiate \(f(x)=\ln\lvert x\rvert\) on either interval \(x<0\) or \(x>0\). \(f'(x)=1/x\) on each interval.
Which angle unit is required for the standard formula \(d(\sin x)/dx=\cos x\)? Radians.
If the difference quotient grows without bound at \(a\), does \(f'(a)\) exist as a finite real number? No. An infinite slope is not a finite derivative.
If \(y=y(x)\) is differentiable, what is \(\frac{d}{dx}y^3\)? \(3y^2\frac{dy}{dx}\).
Evaluate \(\lim_{x\to0^+}x\ln x\). \(0\), after rewriting the product as \(\ln x/(1/x)\).
For real constant \(k\) and \(a<b\), what is \(\int_a^b k\,dx\)? \(k(b-a)\).
Evaluate \(\int_1^e\frac{\ln x}{x}\,dx\). \(1/2\).
Find \(\int\frac{x}{x^2+5}\,dx\). \(\frac12\ln(x^2+5)+C\).
What is \(\frac{d}{dx}\arccos x\) for \(-1<x<1\)? \(-1/\sqrt{1-x^2}\).
For continuous \(f\) and differentiable \(g,h\), what is \(\frac{d}{dx}\int_{h(x)}^{g(x)}f(t)\,dt\) where defined? \(f(g(x))g'(x)-f(h(x))h'(x)\).
For a partition \(a=x_0<\cdots<x_n=b\) and sample points \(x_i^*\in[x_{i-1},x_i]\), what sum approximates the signed area? \(\sum_{i=1}^{n}f(x_i^*)(x_i-x_{i-1})\).
What makes a discontinuity at \(a\) removable? The finite limit at \(a\) exists, and assigning that limit to \(f(a)\) makes the function continuous there.
On an interval within \((0,1)\) or \((1,\infty)\), find \(\int\frac{1}{x\ln x}\,dx\). \(\ln\lvert\ln x\rvert+C\).
If a flow rate is measured in liters per minute and time in minutes, what units does its time integral have? Liters.
For integrable \(f\) on \([a,b]\), \(a<b\), what is its average value? \(\frac{1}{b-a}\int_a^b f(x)\,dx\).
If \(g(x)\le f(x)\le h(x)\) near \(a\), and both outer functions approach \(L\), what follows for \(f\)? \(\lim_{x\to a}f(x)=L\).
How does a definite integral treat regions where the graph lies below the x-axis? Their geometric areas contribute negatively.
If \(f'(c)=0\), \(f''\) is continuous near \(c\), and \(f''(c)<0\), what occurs at \(c\)? A strict local maximum.
In Newton's method, which point on the tangent line at \(x_n\) determines \(x_{n+1}\)? The tangent line's x-intercept.
With radian input, evaluate \(\lim_{x\to0}(1-\cos x)/x\). \(0\).
For \(-4<x<4\), find \(\int\frac{1}{\sqrt{16-x^2}}\,dx\). \(\arcsin(x/4)+C\).
Find \(\int xe^{-x^2}\,dx\). \(-\frac12e^{-x^2}+C\).
If \(u(1)=2\), \(u'(1)=3\), \(v(1)=4\), and \(v'(1)=-2\), what is \((u/v)'(1)\)? \(1\).
What distinguishes a jump discontinuity from a removable discontinuity? Its finite one-sided limits are unequal, so changing just the point value cannot restore continuity.
Use four equal subintervals to find the midpoint sum for \(f(x)=x^2\) on \([0,2]\). \(21/8\), an approximation to the integral.
For \(f(x)=x^2\) restricted to \(x\ge0\), what is its inverse for nonnegative inputs? \(f^{-1}(x)=\sqrt{x}\).
If a differentiable function has a local extremum at an interior point \(c\), what must \(f'(c)\) equal? \(0\).
What is \(\frac{d}{dx}\arcsin x\) for \(-1<x<1\)? \(1/\sqrt{1-x^2}\).
For \(f(x)=\sin x\), with \(x\) in radians, what is \(f''(x)\)? \(-\sin x\).
Does \(f'(c)=0\) alone guarantee a local extremum at \(c\)? No. For \(f(x)=x^3\), \(f'(0)=0\), but the graph keeps increasing through \(0\).
What is the area enclosed by \(y=x^2\) and \(y=4\)? \(32/3\) square units, using the intersection bounds \(-2\) and \(2\).
If \(s'(t)=3t^2-2\) and \(s(0)=7\), what is \(s(t)\)? \(s(t)=t^3-2t+7\).
Does \(f^{-1}(x)\) normally mean \(1/f(x)\)? No. It denotes the inverse function; the reciprocal is \(1/f(x)\).
Square corners of side \(x\) cm are removed from an \(18\) cm square sheet to fold an open box. Which \(x\in(0,9)\) maximizes volume? \(x=3\) cm; \(V(x)=x(18-2x)^2\) reaches \(432\text{ cm}^3\).
Differentiate \(f(x)=(x^2+2)(3x-1)\) for real \(x\). \(f'(x)=9x^2-2x+6\).
How can a proposed indefinite integral \(F(x)+C\) be checked? Differentiate \(F\) and verify that the result equals the integrand throughout the stated interval.
Differentiate \(f(x)=(2x+1)/(x-4)\) for \(x\ne4\). \(f'(x)=-9/(x-4)^2\).
Evaluate \(\lim_{x\to0}x^2\sin(1/x)\). \(0\), since its absolute value is at most \(x^2\).
For \(f(x)=5x-1\), what choice of \(\delta\) ensures \(\lvert f(x)-9\rvert<\epsilon\) whenever \(0<\lvert x-2\rvert<\delta\)? \(\delta=\epsilon/5\), because \(\lvert f(x)-9\rvert=5\lvert x-2\rvert\).
What is the absolute maximum of \(f(x)=x^2-6x+11\) on \([0,5]\)? \(11\), attained at \(x=0\).
For a quotient of form \(0/0\) or \(\infty/\infty\), what local differentiability assumptions are needed for L'Hôpital's rule? The numerator and denominator are differentiable on the relevant punctured interval, and the denominator's derivative is nonzero there. The derivative quotient must have a finite or infinite limit.
After finding an indefinite integral in \(u=g(x)\), how should an answer requested in \(x\) be finished? Replace \(u\) by \(g(x)\) and include the arbitrary constant.
For \(f(x)=x^3\) on \([0,3]\), which point \(c\in(0,3)\) has derivative equal to the secant slope? \(c=\sqrt{3}\).
If \(f\) is continuous at \(c\) and \(f'\) changes from positive just left of \(c\) to negative just right of \(c\), what occurs at \(c\)? A strict local maximum.
What is the linearization of \(f(x)=\ln x\) at \(a=1\)? \(L(x)=x-1\).
What is the largest area of a rectangle with perimeter \(28\) m? \(49\text{ m}^2\), achieved by a \(7\) m by \(7\) m square.
For constant \(a>0\), \(a\ne1\), what is \(\int a^x\,dx\)? \(\frac{a^x}{\ln a}+C\).
With \(x\) in radians and \(\cos x\ne0\), what is \(\frac{d}{dx}\sec x\)? \(\sec x\tan x\).
Differentiate \(f(x)=\ln(2x^2+1)\) for real \(x\). \(f'(x)=\frac{4x}{2x^2+1}\).
For \(f(x)=x^2+3x\) and \(h\ne0\), simplify \((f(x+h)-f(x))/h\). \(2x+h+3\).
Find \(\int(8x^3-6x+5)\,dx\). \(2x^4-3x^2+5x+C\).
Differentiate \(f(x)=7/x\) for \(x\ne0\). \(f'(x)=-7/x^2\).
For \(f(x)=x^2-7\) and \(x_0=3\), what is the first Newton iterate \(x_1\)? \(8/3\).
If continuous boundaries satisfy \(x=R(y)\ge L(y)\) for \(c\le y\le d\), what integral gives the enclosed area? \(\int_c^d(R(y)-L(y))\,dy\).
Why is \(x=0\) not an inflection point of \(f(x)=x^4\), although \(f''(0)=0\)? \(f''(x)=12x^2\) is positive on both sides of \(0\), so concavity does not change.
Evaluate \(\int_{-1}^{2}x^3\,dx\). \(15/4\).
For \(a>0\), \(a\ne1\), and \(x>0\), what is \(\frac{d}{dx}\log_a x\)? \(\frac{1}{x\ln a}\).
If \(V(t)\) is measured in liters and \(t\) in minutes, what are the units of \(V'(t)\)? Liters per minute.
For \(\int_0^1 e^{-2x}\,dx\), substitution \(u=-2x\) gives which u-bounds? Lower bound \(0\), upper bound \(-2\). The factor \(dx=-du/2\) also remains.
How does \(\int_b^a f(x)\,dx\) relate to \(\int_a^b f(x)\,dx\)? \(\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx\).
What is the real domain of \(f(x)=\ln(x-4)\)? \(x>4\).
Evaluate \(\lim_{x\to\infty}\frac{x^2}{e^x}\). \(0\).
Find \(\int\frac{1}{9+x^2}\,dx\). \(\frac13\arctan(x/3)+C\).
What conditions on \(f\) allow Rolle's theorem on \([a,b]\), where \(a<b\)? \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a)=f(b)\).
Where is a rational function continuous? At every point where its denominator is nonzero.
What does it mean for \(f(c)\) to be a local minimum at an interior point \(c\)? For some open interval about \(c\), \(f(c)\le f(x)\) for every domain point \(x\) in that interval.
For an integrable \(f\), what is \(\int_a^a f(x)\,dx\)? \(0\).
Why is solving the objective's derivative equal to zero not enough to finish a constrained optimization problem? The candidates must satisfy the constraints and be compared with relevant endpoints or limiting behavior.
Let \(f(x)=2x+k\) for \(x<3\), and \(f(x)=x^2\) for \(x\ge3\). Which \(k\) makes \(f\) continuous at \(3\)? \(k=3\), since \(6+k=9\).
If \(f(x)\to L\) and \(g(x)\to M\) as \(x\to a\), what extra condition permits the quotient law \(f/g\to L/M\)? \(M\ne0\).
Why does Rolle's theorem not apply to \(f(x)=\lvert x\rvert\) on \([-1,1]\), despite equal endpoint values? The function is not differentiable at the interior point \(0\).
Differentiate \(f(x)=4x^5-3x^2+7\) for real \(x\). \(f'(x)=20x^4-6x\).
Under L'Hôpital's rule's hypotheses, which quotient replaces \(f(x)/g(x)\) for evaluating the limit? \(f'(x)/g'(x)\); this is not the quotient-rule derivative of \(f/g\).
Evaluate \(\int_0^1 e^{2x}\,dx\). \((e^2-1)/2\).
A particle has velocity \(v(t)=2t-4\) m/s for \(0\le t\le5\) s. What is its displacement? \(5\) m.
A square's side is \(8\) meters and is increasing at \(3\) meters per second. What is its area rate at that instant? \(48\) square meters per second.
Evaluate \(\lim_{x\to-3}(x^2+4x+3)/(x+3)\). \(-2\).
How can area between continuous curves be computed if the upper curve changes at intersections? Split the interval at those intersections and integrate upper minus lower on each part.
For \(-1<x<1\), find \(\int\frac{1}{\sqrt{1-x^2}}\,dx\). \(\arcsin x+C\).
For \(x>-1/2\), find \(\int\sqrt{2x+1}\,dx\). \(\frac13(2x+1)^{3/2}+C\).
Use four equal subintervals to find the right-endpoint sum for \(f(x)=x^2\) on \([0,2]\). \(15/4\), an approximation to the integral.
What does continuity on a closed, bounded interval guarantee about Riemann integrability? The function is Riemann integrable on that interval.
In an optimization model, what is the objective function? The quantity to be maximized or minimized, expressed in terms of the chosen variables.
Find \(\int 2x(x^2+4)^3\,dx\). \(\frac{(x^2+4)^4}{4}+C\).
What is the maximum area of a rectangular pen using \(60\) m of fencing on three sides against a wall? \(450\text{ m}^2\), with width \(15\) m and length \(30\) m.
What is the tangent line to \(x^2+xy+y^2=7\) at \((1,2)\)? \(y-2=-\frac45(x-1)\).
Let \(f(x)=(x^2-49)/(x-7)\) for \(x\ne7\). What value of \(f(7)\) makes \(f\) continuous at \(7\)? \(f(7)=14\).
Using linearization of \(\sqrt{x}\) at \(25\), estimate \(\sqrt{25.4}\). Approximately \(5.04\).
Evaluate \(\lim_{x\to\infty}\frac{5x^2-2x+1}{2x^2+7}\). \(5/2\).
Find \(\int\frac{1}{1+x^2}\,dx\). \(\arctan x+C\).
In the definition of \(\lim_{x\to a}f(x)=L\), what must exist for every \(\epsilon>0\)? A \(\delta>0\) such that every domain point with \(0<\lvert x-a\rvert<\delta\) satisfies \(\lvert f(x)-L\rvert<\epsilon\).
For an interior domain point \(a\), what equation characterizes continuity when \(f(a)\) and the limit exist? \(\lim_{x\to a}f(x)=f(a)\).
Where is the local maximum of \(f(x)=x^3-12x\), and what is its value? At \(x=-2\), with \(f(-2)=16\). The derivative changes from positive to negative.
For \(s(t)=2t^3-t^2\) meters, what is acceleration at \(t=1\) second? \(10\) meters per second squared.
For \(f(x)=x^4-5x^2\), find \(f''(x)\). \(12x^2-10\).
Differentiate \(f(x)=(3x^2-1)^4\) for real \(x\). \(f'(x)=24x(3x^2-1)^3\).
If \(f(x)\le g(x)\) on \([a,b]\), with \(a<b\) and both functions integrable, how do their integrals compare? \(\int_a^b f(x)\,dx\le\int_a^b g(x)\,dx\).
If \(f'(c)=0\), \(f''\) is continuous near \(c\), and \(f''(c)>0\), what occurs at \(c\)? A strict local minimum.
A rectangular pen against a straight wall needs fencing on two widths \(x\) and one length. With \(60\) m of fencing, what is its area as a function of \(x\in(0,30)\)? \(A(x)=x(60-2x)\) square meters.
For \(x\ne0\), find \(\int x^{-3}\,dx\) on an interval avoiding \(0\). \(-\frac{1}{2x^2}+C\).
For \(x>0\), what is the minimum of \(x+36/x\)? \(12\), attained at \(x=6\). The derivative is negative before \(6\) and positive after it.
When the endpoints and integrand are fixed, is a definite integral a number or a family of antiderivatives? A number, if the integral exists.
In radians, evaluate \(\lim_{x\to0}\frac{\sin(3x)}{\sin(7x)}\). \(3/7\).
When \(u=g(x)\) in an integral, what differential replaces \(g'(x)\,dx\)? \(du\).
With \(x\) in radians and \(\sin x\ne0\), what is \(\frac{d}{dx}\csc x\)? \(-\csc x\cot x\).
Why can the ordinary FTC evaluation rule not be used on \(\int_{-1}^{1}1/x\,dx\)? The integrand is undefined at \(0\) and is not continuous on the interval. This is an improper integral, whose two one-sided pieces do not converge.
If \(\lim_{x\to2^-}f(x)=3\) and \(\lim_{x\to2^+}f(x)=5\), what is \(\lim_{x\to2}f(x)\)? It does not exist because the one-sided limits differ.
Find \(\int x\sqrt{x^2+9}\,dx\). \(\frac{(x^2+9)^{3/2}}{3}+C\).
In an epsilon-delta proof at a fixed \(a\), may the chosen \(\delta\) depend on the particular varying \(x\)? No. It must work for every allowed \(x\) in the specified neighborhood; it may depend on \(\epsilon\) and the fixed data.
If price is \(p=50-q\) and \(0\le q\le50\), which quantity maximizes revenue \(R=q(50-q)\)? \(q=25\), giving maximum revenue \(625\) in the model's monetary units.
What is the slope of \(y=\arctan x\) at \(x=1\)? \(1/2\).
Evaluate \(\lim_{h\to0}(\sqrt{16+h}-4)/h\). \(1/8\).
If \(A(x)=\int_a^x f(t)\,dt\), what is \(A(a)\)? \(0\).
For a sphere with \(V=4\pi r^3/3\), radius \(2\) cm, and \(V'=-16\pi\) cubic centimeters per second, what is \(r'\)? \(-1\) centimeter per second.
For \(s(t)=t^3-6t\) meters, what is velocity at \(t=2\) seconds? \(6\) meters per second.
For \(f(x)=2x^5-x^3\), find \(f^{(3)}(x)\). \(120x^2-6\).
For continuous \(f\ge g\) on \([a,b]\), where \(a<b\), what integral gives the area between their graphs? \(\int_a^b(f(x)-g(x))\,dx\).
Does obtaining \(0/0\) by substitution determine the value of a limit? No. It is an indeterminate form and requires further analysis.
For a fixed base \(a>0\), what is \(\frac{d}{dx}a^x\)? \(a^x\ln a\).
What equation makes \(F\) an antiderivative of \(f\) on an interval \(I\)? \(F'(x)=f(x)\) for every \(x\in I\).
For real-valued \(f\) and \(g\), which inputs belong to the domain of \(f\circ g\)? Inputs in the domain of \(g\) whose outputs \(g(x)\) lie in the domain of \(f\).
Why does an indefinite integral on an interval include \(+C\)? Differentiation removes an additive constant; all antiderivatives on that interval differ by a constant.
If \(f'(x)>0\) throughout an interval, how does \(f\) vary there? It is strictly increasing.
If \(f'(x)=0\) for every \(x\) in an open interval \(I\), what follows about \(f\) on \(I\)? \(f\) is constant on \(I\).
If \(f\) is odd and integrable on \([-a,a]\), where \(a>0\), what is \(\int_{-a}^{a}f(x)\,dx\)? \(0\).
What is the average value of \(f(x)=x^2\) on \([1,4]\)? \(7\).
What limit condition makes \(y=L\) a horizontal asymptote of \(f\)? \(\lim_{x\to\infty}f(x)=L\) or \(\lim_{x\to-\infty}f(x)=L\), with \(L\) finite.
When geometric quantities change with time, with respect to which variable should their relation be differentiated to link their rates? Time, applying the chain rule to each changing quantity.
Evaluate \(\int_1^3(2x+1)\,dx\). \(10\).
Differentiate \(f(x)=x^2e^x\) for real \(x\). \(f'(x)=e^x(x^2+2x)\).
If \(f''>0\) throughout an interval, what is the graph's concavity there? Concave up; the slopes \(f'\) increase.
Where is a real polynomial continuous? At every real number.
For a continuous function on \([a,b]\), which values should be compared to find absolute extrema when its interior critical numbers are known? Evaluate \(f\) at the endpoints and at every interior critical number, then compare the values.
If \(f''>0\) throughout an interval containing \(a\), is the tangent-line approximation at \(a\) above or below \(f\) elsewhere in that interval? Below \(f\); a concave-up graph lies above its tangent line.
Why does using \(F+C\) instead of \(F\) leave a definite integral's value unchanged? The same \(C\) occurs at both endpoints and cancels in the subtraction.
In radians, evaluate \(\int_0^{\pi/2}\sin x\cos x\,dx\). \(1/2\).
What is a critical number of a function \(f\)? An interior point \(c\) of its domain where \(f'(c)=0\) or \(f'(c)\) does not exist.
Can the domain of \(f'\) be smaller than the domain of \(f\)? Yes. For example, \(\sqrt{x}\) is defined at \(0\), but has no finite derivative there.
On an interval where \(\cos(2x)\ne0\), find \(\int\sec^2(2x)\,dx\). \(\frac12\tan(2x)+C\).
For continuous velocity \(v(t)\) on \([a,b]\), \(a<b\), what integral gives displacement? \(\int_a^b v(t)\,dt\).
Evaluate \(\lim_{x\to5}(x^2-25)/(x-5)\). \(10\), after canceling \(x-5\) for \(x\ne5\).
Which two quotient forms directly qualify for L'Hôpital's rule when its derivative hypotheses also hold? \(0/0\) and \(\infty/\infty\), including signed infinite limits.
For \(f(x)=x^2+1\) and \(g(x)=3x-2\), what is \(f(g(x))\)? \((3x-2)^2+1\).
Differentiate \(f(x)=x/(x^2+4)\) for real \(x\). \(f'(x)=\frac{4-x^2}{(x^2+4)^2}\).
If \(m\le f(x)\le M\) on \([a,b]\), where \(a<b\) and \(f\) is integrable, what bounds hold for its integral? \(m(b-a)\le\int_a^b f(x)\,dx\le M(b-a)\).
If \(f'(c)=0\) and \(f''(c)=0\), what does the second derivative test decide? Nothing by itself; another test is needed.
Evaluate \(\int_2^8\frac{1}{x}\,dx\). \(\ln4\).
Evaluate \(\lim_{x\to4^-}1/(x-4)\). \(-\infty\).
If \(f\) is continuous at \(c\) and \(f'\) changes from negative just left of \(c\) to positive just right of \(c\), what occurs at \(c\)? A strict local minimum.
What slope equality does the Mean Value Theorem guarantee on \([a,b]\)? For at least one \(c\in(a,b)\), \(f'(c)=\frac{f(b)-f(a)}{b-a}\).
What change in a continuous graph characterizes an inflection point? The graph changes concavity at that point.
What is \(\frac{d}{dx}\arctan x\) for real \(x\)? \(1/(1+x^2)\).
What one-sided behavior is sufficient for \(x=a\) to be a vertical asymptote of \(f\)? At least one of the one-sided limits as \(x\to a\) is \(+\infty\) or \(-\infty\).
A sphere has radius \(2\) cm and volume increasing at \(12\pi\) cubic centimeters per second. With \(V=4\pi r^3/3\), what is \(dr/dt\)? \(3/4\) centimeter per second.
Evaluate \(\lim_{x\to0}\frac{e^{5x}-1}{x}\). \(5\).
A rigid \(13\)-meter ladder rests against a vertical wall. Its base is \(5\) meters from the wall and moves away at \(3\) m/s. How fast is the top's height changing? \(-5/4\) m/s; the top moves downward.
For \(f(x)=x^2\) on \([1,5]\), which point \(c\in(1,5)\) has derivative equal to the secant slope? \(c=3\).
On which open intervals is \(f(x)=x^3-12x\) increasing? \((-\infty,-2)\) and \((2,\infty)\), because \(f'(x)=3(x^2-4)>0\) there.
If \(f\) is even and integrable on \([-a,a]\), where \(a>0\), how can \(\int_{-a}^{a}f(x)\,dx\) be simplified? \(2\int_0^a f(x)\,dx\).
With radian input, evaluate \(\lim_{x\to0}\sin(5x)/x\). \(5\).
Differentiate \(f(x)=\ln x/x\) for \(x>0\). \(f'(x)=\frac{1-\ln x}{x^2}\).
A circle has radius \(3\) cm increasing at \(2\) cm/s. With \(A=\pi r^2\), what is \(dA/dt\) at that instant? \(12\pi\) square centimeters per second.
On an interval where \(\cos x\ne0\), find \(\int\sec^2x\,dx\). \(\tan x+C\).
Must an antiderivative of \(1/x\) use the same additive constant on \((-\infty,0)\) and \((0,\infty)\)? No. The two disconnected intervals may have independent constants.
In radians, find \(\int\cos x\,dx\). \(\sin x+C\).
Why does the limit definition require \(0<\lvert x-a\rvert\)? It excludes \(x=a\), so the function's value at that point does not control the limit.
If \(y=y(x)\) is differentiable, what is \(\frac{d}{dx}(xy)\)? \(y+x\frac{dy}{dx}\).
Evaluate \(\lim_{x\to0}\frac{e^x-1-x}{x^2}\). \(1/2\).
For integrable \(f\) and \(a<c<b\), how is \(\int_a^b f\) split at \(c\)? \(\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx\).
Differentiate \(f(x)=\arctan(3x)\) for real \(x\). \(f'(x)=3/(1+9x^2)\).
For \(a\ne b\), what is the slope of the secant through \((a,f(a))\) and \((b,f(b))\)? \(\frac{f(b)-f(a)}{b-a}\).
For differentiable \(y=f(x)\), how is the differential \(dy\) related to \(dx\)? \(dy=f'(x)\,dx\).
Evaluate \(\lim_{x\to-\infty}\frac{4x-1}{x^2+3}\). \(0\).
If both one-sided limits at \(a\) are \(6\) but \(f(a)\) is undefined, what is \(\lim_{x\to a}f(x)\)? \(6\).
For integrable \(f,g\) and constants \(r,s\), how does \(\int_a^b(rf+sg)\,dx\) expand? \(r\int_a^b f(x)\,dx+s\int_a^b g(x)\,dx\).
How you study it in Repeto
- Install the deck free. It lands in your library on the web, iPhone, iPad and Mac.
- Answer from memory. See the front, recall the answer, then flip and rate how it went.
- Review at the right time. FSRS spaced repetition brings each card back just before you would forget it, so a few minutes a day is enough.
Sources and license
Original Repeto-authored study cards and diagrams, based on the documented public educational sources; not endorsed by their publishers.