Calculus Problems and Solutions
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Determine $(g\circ f)(x)$ when $f(x)=-2\,x^2-4$ and $g(x)=-x$.
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Determine the derivative of $f(g(x))$ where $f(x)=\sin(x)$ and $g(x)=x^2+4$.
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Determine the average value of $f(x)=x^2$ between $0$ and $2$.
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If $\int_0^1f(x)\,dx=16$ and $\int_0^1g(x)\,dx=3.5$, then determine $\int_0^14\cdot f(x)-8\cdot g(x)\,dx$.
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Determine the antiderivative of $f(x)=x$.
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Determine \[ \int_0^ax\,dx. \]
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Determine \[ \int_1^2\frac{1}{x^3}\,dx. \]
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Determine \[ \int_0^{10}x^4-10x^3+5000\,dx. \]
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Determine \[ \int_3^{10}2x^2-x+15\,dx. \]
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Determine the antiderivative of $f(x)=1/x^3$.
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Determine the antiderivative of $f(x)=x^4-10x^3+5000$.
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Determine the antiderivative of $f(x)=2x^2-x+15$.
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Use the integral notation to write the area of $f(x)=5x^3−18x^2−10$ between $−1$ and $3$.
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Use the integral notation to write the indefinite integral of $x^2+5$.
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Determine the inflection points of $f(x)=x^4-10x^3$.
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Determine the concavity of $f(x)=x^4-10x^3$.
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Determine the maxima and minima of $f(x)=-(x+7)^2+8$.
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Determine the maxima and minima of the function $f(x)=x^4-10x^3$.
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Determine the critical points of $f(x)=(x+2)^2$.
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Determine whether $f(x)=-8x+73$ is increasing or decreasing.
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Determine whether $f(x)=5x-3$ is increasing or decreasing.
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Show that $f(x)=x^2$ is not a decreasing function between $a=-10$ and $b=2$ by evaluating $f\,'(1)$.
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Determine the second derivative of $f(x)=-5x^3+17x^2-5x+3$.
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Determine the average speed of a car between an hour and a half and a half hour after it starts a trip when its distance-time function is $d(t)=t^3-15t^2+75t$.
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Determine the instantaneous speed of a car three quarters of an hour after starting on a trip where its distance-time function is $d(t)=t^3-15t^2+75t$.
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Determine $(g\circ f)\,'(x)$ where $f(x)=3x^2+5x-2$ and $g(x)=4x+3$.
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Determine $(f\circ g)\,'(x)$ where $f(x)=3x^2+5x-2$ and $g(x)=4x+3$.
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Determine the derivative of $(x^2-1)^{50}$.
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Use the quotient rule to find the derivative of $f(x)/g(x)$ where \begin{eqnarray} f(x)&=&5x^6-16x^5-4x^4\\ &+&72x^3-3x^2-2x+18 \end{eqnarray} and $g(x)=42x^2+32x+7$.
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Use the quotient rule to find the derivative of $f(x)/g(x)$ where $f(x)=32x^2-23x+6$ and $g(x)=4x^2+7x+6$.
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Use the quotient rule to find the derivative of $f(x)/g(x)$ where $f(x)=4x^2+5x-2$ and $g(x)=-3x-7$.
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Use the product rule to find the derivative of $f(x)g(x)$ where $f(x)=32x^2-23x+6$ and $g(x)=4x^2+7x+6$.
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Use the product rule to find the derivative of $f(x)g(x)$ where $f(x)=5x^6-16x^5-4x^4+72x^3-3x^2-2x+18$ and $g(x)=42x^2+32x+7$.
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Use the product rule to find the derivative of $f(x)g(x)$ where $f(x)=4x^2+5x-2$ and $g(x)=-3x-7$.
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Prove that $L(P)\leq R(P)\leq U(P)$.
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Prove that if \begin{eqnarray} a_1&\leq& b_1\leq c_1,\\ a_2&\leq& b_2\leq c_2,\\ &\vdots&\\ a_N&\leq& b_N\leq c_N, \end{eqnarray} then \[ \sum_{i=1}^Na_i\leq \sum_{i=1}^Nb_i\leq \sum_{i=1}^Nc_i. \]
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Prove that if $a_1\leq b_1\leq c_1$ and $a_2\leq b_2\leq c_2$, then $a_1+a_2\leq b_1+b_2\leq c_1+c_2$.
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Prove that if $a_1\leq b_1$ and $a_2\leq b_2$, then $a_1+a_2\leq b_1+b_2$.
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Prove that continuous functions are integrable.
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Determine the definite integral of any polynomial.
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Prove the Fundamental Theorem of Integral Calculus
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Prove that if $f(x)$ is continuous on the closed interval $[a,b]$, then $f(x)$ is uniformly continuous on $[a,b]$.
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Prove that $f(x)=x^2$ is not uniformly continuous on the entire $x$-axis.
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Prove that if $f(x)$ is uniformly continuous on a set $U$ and $a$ is inside $U$, then $f(x)$ is continuous at $a$.
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Show that all $x$ satisfying $0\lt |x-a|\lt\delta$ is equivalent to the union $(a-\delta,a)\cup (a,a+\delta)$ where $a$ is any real number and $\delta\gt 0$.
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Find the union and intersection of $A=\{-10,0,-101,3,5,19\}$ and $B=\{-101,2,7,5,10,100\}$.
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What is the union of the citizens of England, Scotland, Wales and Northern Ireland?
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Find the union and the intersection of the sets $A=\{0,2,4,6,8,\cdots\}$ and $B=\{1,3,5,7,9,\cdots\}$.
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Find the union and intersection of $A=\{1,2,3,4,5\}$ and $B=\{2,7,5,10,100\}$.
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Find the union and intersection of the following two sets: $H=\{dog,cat,hamster,goldfish\}$ and $F=\{cow,sheep,dog,horse,cat,chicken\}$.
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