# Derivatives Practice Quiz

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This Lesson:
What is a Derivative? |
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Derivative By Definition |

## Practice Problems

**Problem 1)**

*Identify the tangent line and the secant line in the graph below*

Answer |

Tangent= because the line has the same slope as the curve at that one point.

Secant= because the slope of this line is found using 2 points on the curve.

**Problem 2)**

*Sketch the secant line between the point a and point b. On the same graph below sketch the tangent line at point a.*

Answer |

Grey Line= tangent line

Green Line= Secant

**Problem 3)**

*1) Find the average rate of change between the points (-1,6) and (5,3)*

Answer |

The average rate of change can be found by means of the slope formula.

$ slope = \frac{ \Delta y }{ \Delta x} = \frac{3-6}{5--1}= \frac{-3}{6} = -\frac{1}{2} $

**Problem 4)**

*1) Find the equation of the secant line between point a and point b in the graph below.*

Answer |

Slope: $ slope = \frac{3-1}{9-1}= \frac{2}{8} = \frac{1}{4} $

Equation in point slope form: $ (y-3) = \frac{1}{4} (x -9) $

Alternate equation in point slope form: $ (y-1) = \frac{1}{4} (x -1) $

Equation in slope intercept form: $ y= \frac{1}{4} +\frac{3}{4} $

**Problem 5)**

- A)
*Sketch the graph of $$y= (x-1)^2 +2 $$*

Answer |

- B)
Plot the point x = -1 and label it "a"- C)
Plot the point x= 1 and label it "b"- D) Find the equation of the secant line between points "a" and points "b"

Answers for B,C, and D

Slope: $ $slope = \frac{2-4}{1--1}= \frac{-2}{2} = -1$$

Equation: (y-2)=-1(x-1) or y=-x+3

Graph Secant Line

**Problem 7)**Given function y = x

^{3}, and the point (-1,-1). Starting with the given point which x-value will produce a secant line with the greatest rate of change.

- x =1
- x = 0
- x = -2
- x = 2

Answer |

Points Slope Choice a (1,1)

Given point (-1,-1)$\frac{-1-1}{-1-1}= \frac{-2}{-2} = 1$ Choice b (0,0)

Given point (-1,-1)$\frac{-1-0}{-1-0}= \frac{-1}{-1} = 1$ Choice c (-2,8)

Given point (-1,-1)$\frac{-1-8}{-1--2}= \frac{-9}{1} = -9$ Choice c (2,8)

Given point (-1,-1)$\frac{-1-8}{-1-2}= \frac{-9}{-3} = 3$

This secant's slope is the greatest and therefore is the answer

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Derivative By Definition

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