# MATH221 Study Guide - Final Guide: Farad, Product Rule, Natural Logarithm

Find the slope at the following points using the equation y=x^2

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Slope of a tangent line at point P= (change in y)/(change in x)

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1.2: The Slope of a Curve at a Point

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(-.4, .16)

a.

(-2, 4)

b.

(-1.5, 2.25)

c.

The derivative of f(x) is denoted by f'(x)

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If f(x)=mx+b, then f'(x)=m

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If f(x)=x^r, f(x)=r*x^r-1

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The derivative is also the rate of change for an equation

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To find the equation for the tangent line: y-f(a)=f'(a)(x-a)

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The derivative is the slope of a line at a given point

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The secant line is a line that passes through 2 points of f(x), whereas

the derivative only passes through 1 points

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The slope of the secant line= [f(x+h)-f(x)] / [(x+h)-x]

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f(x)=3x+7

1.

Find the derivative:

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1.3: The Derivative

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f(x)= (3x/4)-2

2.

f(x)=x^7

3.

f(x)=x^2/3

4.

f(x)= 1/(x)^5/2

5.

Lim g(x)=L

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The limit of g(x) is L as it approaches a, if the values of g(x)

approach L as x approaches a

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Limit Rules:

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x--->a

1.4: Limits and the Derivative:

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Final Exam Review

Monday, December 3, 2018

5:09 PM

FINAL EXAM REVIEW Page 1

Limit definition of the Derivative

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Write the difference quotient (above)

1.

Simplify the difference quotient

2.

Find the limit as h-->0. This limit is f'(a)

3.

Calculate f'(a) using limits:

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f(x)=3x+1

1.

Use limits to compute f'(x) for the given function:

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f(x)=x+(1/x)

2.

f(x)=x/[x+1]

3.

1.

2.

Determine the function f(x) and the value of a for the following

equations

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1.

2.

Compute the following limits:

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3.

FINAL EXAM REVIEW Page 2

3.

1.6: Some Rules for Differentiation

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Rules:

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Differentiate:

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1.

Solve:

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1.8 The derivative as a Rate of Change

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The rate of change of f(x) from a to b= [f(b)-f(a)] / [b-a]

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The derivative f'(a) measures the instantaneous rate of change of f(x)

at x=a

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s(t)= position

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v(t)= velocity

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s'(t)=v(t)

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v'(t)=a(t)

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s''(t)=a(t)

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a(t)= acceleration

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To approximate a change with the derivative: f(a+h)-f(a)~f'(a)*h

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Practice:

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FINAL EXAM REVIEW Page 3

## Document Summary

1. 2: the slope of a curve at a point. Slope of a tangent line at point p= (change in y)/(change in x) Find the slope at the following points using the equation y=x^2 a. (-. 4, . 16) b. (-2, 4) c. (-1. 5, 2. 25) The derivative is the slope of a line at a given point. The derivative of f(x) is denoted by f"(x) The derivative is also the rate of change for an equation. To find the equation for the tangent line: y-f(a)=f"(a)(x-a) The secant line is a line that passes through 2 points of f(x), whereas the derivative only passes through 1 points. The slope of the secant line= [f(x+h)-f(x)] / [(x+h)-x] The limit of g(x) is l as it approaches a, if the values of g(x) approach l as x approaches a. Use limits to compute f"(x) for the given function: Determine the function f(x) and the value of a for the following equations.