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Calculus 1: test 3

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This is a set of sample tests for Calculus 1: course.

1. Sketch the graph of a differentiable function $f$ that is continuous on $[-\infty,6]$ and has an absolute maximum at $-1$, a local maximum at $2$, a local minimum at $5$, an inflection point at $4$, and a horizontal asymptote $y=1$.

2. The velocity of the object at time t is given by $v(t)=1+3t^2$. If at time $t=1$ the object is at position $x=4$, where is it at time $t=0$?

3. Find all antiderivatives of the function $f(x)=x^{\pi}+e^{x}-1/x+e$.

4. (a) Analyze the first and second derivatives of the function $f(x)=x^4-2x^2$. (b) Use part (a) to sketch its graph of $f$.

5. (a) Finish the statement "If $h'(x)=0$ for all $x \in (a,b)$ then...". (b) Finish the statement "If $f'(x)=g'(x)$ for all $x \in (a,b)$ then...". (c) Use part (a) to prove part (b).

6. Find two numbers $x,y$ whose sum is $2$ and whose product is a maximum.

7. Set up but do not solve the optimization problem for the following situation: "Among all rectangles inscribed in a circle of radius $1$, find the one with the largest area".