4 Derivative-Based Analysis Online Quiz Questions

True or false: Geometrically, the derivative at a point is the slope of the tangent line to the graph at that point.

What is $\frac{d}{dx}(7)$?

If $y=x^2e^x$, which expression is $y'$?

Which procedure correctly finds the global maximum and minimum of a continuous function on $[a,b]$?

Select all statements that are correct about critical numbers and local extrema.

Select all statements that correctly describe concavity or inflection points.

True or false: A marginal cost is generally a local approximation of the additional cost, rather than necessarily the exact cost of producing the next unit.

True or false: If $f'(c)=0$ and $f''(c)=0$, the second derivative test proves that $c$ is a local maximum.

For $C(q)=500+12q+0.02q^2$, what is the marginal cost $C'(100)$? Enter the numerical value in dollars per additional unit.

For $f(x)=x^3-3x^2+2$, what is the value of $f'(3)$? Enter the numerical value.

A population model satisfies $P'(5)=-120$ people per month. Write a responsible interpretation of this derivative. Include the direction and approximate rate of change, the relevant time, the units, and why the result should not be treated as an exact constant change for all later months.

Which statement best describes what the derivative of a function measures at a particular input?

Select the derivative of $f(x)=3x^4-2x^2+7$.

Using the product rule, what is the derivative of $y=x^2e^x$?

Find $\frac{d}{dx}(\sqrt{x})$. Enter the simplified derivative expression.

For the cost model $C(q)=500+12q+0.02q^2$, what is the marginal cost at $q=100$? Enter the value in dollars per unit.

For $y=\frac{x^2}{x+1}$, where $x\ne -1$, which expression is $y'$?

For $f(x)=x^3$, what does the first derivative test conclude about the critical point at $x=0$?