Calculus Problems and Solutions
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Find the critical points of $f(x)=x^3$.
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Assume $f(x)$ is differentiable at $e$. Prove that if $f\,'(x)\lt 0$, then $f(e-k)\gt f(e)\gt f(e+k)$ for all positive $k$ sufficiently small.
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Assume $f(x)$ is differentiable at $e$. Prove that if $f\,'(x)\gt 0$, then $f(e-k)\lt f(e)\lt f(e+k)$ for all positive $k$ sufficiently small.
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Prove the Critical Point Theorem
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Prove Theorem 2 in the lesson 'Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions'.
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Prove Theorem 1 in the lesson 'Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions'.
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Prove that $f(x)=x^2$ decreases on the interval $(-\infty,0]$ and increases on the interval $[0,\infty)$.
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Prove that \[\lim_{h\to 0}\,\frac{h^3}{h}=0.\]
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The derivative of $x^3$.
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Prove that $f(x)=x^3$ increases everywhere.
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Prove the Mean-Value Theorem.
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Prove Rolle's Theorem.
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Set of all real numbers whose square is greater than $27$.
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The set of all real numbers whose square is $4$.
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Set of all integers greater than $100$.
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Marginal Cost
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Find the Marginal Profit
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Revenue of Sand Blasting Bolts
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Marginal Analysis of a Demand Function
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Marginal Analysis of Advertising
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Marginal Analysis of a Snow Blower
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Marginal Analysis of Office Complex
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Marginal Analysis of Air Conditioner Units
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Mutual Fund Value Growth
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Determine the rate of change of the surface area of a cylinder with respect to its radius.
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Determine the rate of change of the area of a square with respect to the length of its sides.
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Determine the rate of change of the area of an equilateral triangle with respect to the length of a side.
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Supply and Demand of Wireless External Hard Drives
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Solve the Quadratic Equation $3x^2+5x-2=0$.
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Supply and Demand of Roof Shingles
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ACME Soap Company Sales Revenue Growth
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Automotive Repair Shop Growth
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Solve the Quadratic Equation.
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Suppose that your distance from home on a Sunday afternoon drive was given by the formula $D(t)=-7.7t^3+3.1t^2+24.6t$ where $D$ is the distance in miles and $t$ is the number of hours driven. Determine your speed during your two hour trip.
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Find the average rate of change of $f(x)=x^2$ between $-2$ and $2$. Find the instantaneous rate of change at both of those points.
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Determine the rate of change of the volume of a sphere with respect to its radius.
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Determine the rate of change of the surface area of a sphere with respect to its radius.
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Determine the rate of change of the circumference of a circle with respect to its radius.
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Determine the rate of change of the area of a circle with respect to its radius.
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Prove that if $f(x)$ has a greatest lower bound $L$, then for every $\epsilon\gt 0$ there exists an $\bar{x}$ in the domain of $f(x)$ such that $|f(\bar{x})-L|\lt\epsilon$.
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Prove that if $f(x)$ is continuous on $[a,b]$, then $f(x)$ takes on a minimum value on $[a,b]$.
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Prove the Extreme-Value Theorem.
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Prove that for every $\epsilon\gt 0$ there exists a natural number $\bar{n}$ such that for all natural numbers $n\geq\bar{n}$, $1/n\lt\epsilon$.
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Prove that the set of natural numbers is unbounded.
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Prove that if $f(x)$ has a least upper bound $M$ then for every $\epsilon\gt 0$ there exists an $\bar{x}$ in the domain of $f(x)$ such that $|f(\bar{x})-M|\lt\epsilon$.
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Prove that if $f(x)$ is continuous on $[a,b]$, then $f(x)$ takes on a maximum value on $[a,b]$.
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Prove that if $f(x)$ is continuous and $K$ is a constant, then $f(x)+K$ is a continuous function.
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Prove the Intermediate-Value Theorem
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$ or $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.
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