The Derivative and Its Definition: math test, College – download the PDF

1 Definition of the derivative / 4 pts

Let \(f(x)=2x^2-x\).

1. Simplify the difference quotient \(\dfrac{f(1+h)-f(1)}{h}\).
2. Use it to find \(f'(1)\).
3. Check your answer with the power rule and sum rule.

2 Differentiating / 4 pts

Differentiate each function.

1. \(g(x)=5x^4-2x^3+x-8\)
2. \(h(x)=3\sqrt{x}+\dfrac{4}{x^2}\)
3. \(k(x)=x(x+4)\)

3 Tangent lines of a cubic / 4 pts

Let \(f(x)=x^3-2x^2+1\).

1. Find the tangent line at \(x=2\).
2. Find the points of the graph where the tangent is horizontal.

4 Filling a tank / 3 pts

The volume of water in a tank is \(V(t)=0.5t^2+4t\) gallons after \(t\) minutes.

1. Find the average rate of change between \(t=2\) and \(t=6\).
2. Find \(V'(t)\), then \(V'(2)\) and \(V'(6)\).
3. What do the results say about the filling speed?

5 A differentiable piecewise function / 3 pts

Let \(f(x)=x^2+1\) for \(x\le 2\) and \(f(x)=mx+n\) for \(x>2\). Find \(m\) and \(n\) so that \(f\) is differentiable at \(2\).

6 Continuity and differentiability / 2 pts

True or false? Justify. (a) A function differentiable at \(a\) is continuous at \(a\). (b) A function continuous at \(a\) is differentiable at \(a\).