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

1 Average rate of change ★★★

Let \(f(x)=x^2+1\). Compute the average rate of change of \(f\) from \(x=1\) to \(x=4\).

2 A linear difference quotient ★★★

Let \(f(x)=5x-2\). Simplify \(\dfrac{f(x+h)-f(x)}{h}\) and find \(f'(x)\).

3 Power rule practice ★★★

Differentiate: (a) \(x^7\); (b) \(x^{10}\); (c) \(6x^4\).

4 A quadratic polynomial ★★★

Find the derivative of \(f(x)=3x^2-8x+11\).

5 Skateboard distance ★★★

A skateboarder’s distance from a ramp is \(s(t)=4t^2\) feet after \(t\) seconds. Find the average velocity between \(t=1\) and \(t=3\), in feet per second and in meters per second (1 ft = 0.3048 m).

6 True or false? ★★★

Decide whether each statement is true or false and justify. (a) \(\dfrac{d}{dx}(12)=12\). (b) \(\dfrac{d}{dx}(5x)=5\). (c) \(\dfrac{d}{dx}(x^4)=4x^4\). (d) If \(f\) is continuous at \(a\), then \(f\) is differentiable at \(a\).

7 Slope of a tangent ★★★

Find the slope of the tangent to \(f(x)=x^2\) at \(x=3\), then write the tangent line.

8 Derivative at a point from the definition ★★★

Let \(f(x)=x^2-4x\). Use the limit definition to find \(f'(2)\).

9 Tangent line to a cubic ★★★

Find the equation of the tangent line to \(f(x)=2x^3-x\) at \(x=1\).

10 Temperature during the day ★★★

The temperature in a greenhouse is \(T(t)=-0.5t^2+6t+50\) degrees Fahrenheit, \(t\) hours after sunrise. (a) Find \(T'(t)\). (b) Interpret \(T'(2)\) and \(T'(8)\). (c) When is the temperature at its peak, and what is it in \(^\circ\)F and in \(^\circ\)C?

11 Horizontal tangents ★★★

Find the points where the graph of \(f(x)=x^3-12x\) has a horizontal tangent.

12 Powers and roots ★★★

Let \(f(x)=\dfrac{1}{x^2}+3\sqrt{x}\). Find \(f'(x)\) and then \(f'(4)\).

13 A corner ★★★

Let \(f(x)=|x-2|\). Show that \(f\) is not differentiable at \(x=2\).

14 Where is the slope 4? ★★★

At which point of the parabola \(y=x^2-6x+1\) is the tangent line of slope \(4\)?

15 Definition applied to a reciprocal ★★★

Use the limit definition to find the derivative of \(f(x)=\dfrac{1}{x}\) for \(x\neq 0\). Then compute \(f'(2)\) and write the tangent line at \(x=2\).

16 Making a piecewise function differentiable ★★★

Let \(f(x)=x^2\) for \(x\le 1\) and \(f(x)=ax+b\) for \(x>1\). Find \(a\) and \(b\) so that \(f\) is differentiable at \(x=1\).

17 Marginal cost ★★★

A workshop’s cost to produce \(q\) units is \(C(q)=0.02q^2+3q+500\) dollars. (a) Find \(C'(q)\). (b) Compute \(C'(100)\) and compare it with the actual extra cost \(C(101)-C(100)\).

18 Tangent lines through an outside point ★★★

Find every tangent line to the parabola \(y=x^2\) that passes through the point \((0,-4)\).

19 Parallel tangents ★★★

Find the tangent lines to \(y=x^3\) that are parallel to the line \(y=12x+5\).

20 A ball thrown upward ★★★

A ball’s height in feet is \(h(t)=-16t^2+96t+10\), \(t\) in seconds. (a) Find the velocity \(v(t)=h'(t)\). (b) Find \(v(1)\) in ft/s and m/s. (c) When does the ball reach its maximum height, and how high is it (in feet and meters)? (d) Find \(v(5)\) and explain its sign.

21 A vertical tangent ★★★

Let \(f(x)=x^{1/3}\) (the cube root). Show that \(f\) is continuous but not differentiable at \(0\), and say what the graph looks like there.