How To Find The Derivative On A Graph

GRAPHING OF FUNCTIONS USING FIRST AND 2d DERIVATIVES

The post-obit problems illustrate detailed graphing of functions of 1 variable using the outset and 2d derivatives. Problems range in difficulty from boilerplate to challenging. If you are going to try these bug before looking at the solutions, you tin avoid common mistakes by carefully labeling critical points, intercepts, and inflection points. In addition, it is important to label the distinct sign charts for the offset and second derivatives in order to avert unnecessary confusion of the following well-known facts and definitions.

Here are instruction for establishing sign charts (number line) for the first and second derivatives. To constitute a sign nautical chart (number lines) for f' , start set f' equal to zero and and then solve for 10 . Marking these x-values underneath the sign chart, and write a naught above each of these ten-values on the sign chart. In improver, marking x-values where the derivative does non be (is non defined). For example, mark those x-values where division by nil occurs in f' . Above these x-values and the sign chart draw a dotted vertical line to indicate that the value of f' does not be at this point. These designated 10-values establish intervals along the sign chart. Side by side, choice points between these designated x-values and substitute them into the equation for f' to decide the sign ( + or - ) for each of these intervals. Beneath each designated x-value, write the respective y-value which is found by using the original equation y = f(x) . These ordered pairs (10, y) will exist a starting point for the graph of f . This completes the sign chart for f' . Plant a sign chart (number line) for f'' in the exact same manner. To avoid overlooking zeroes in the denominators of f' and f'' , it is helpful to rewrite all negative exponents as positive exponents and then carefully dispense and simplify the resulting fractions.

FACTS and DEFINITIONS

$\uparrow$

2. If the first derivative f' is negative (-) , and then the function f is decreasing ( $\downarrow$ ) .

iii. If the 2d derivative f'' is positive (+) , then the part f is concave up ( $\cup$ ) .

4. If the 2nd derivative f'' is negative (-) , then the function f is concave down ( $\cap$ ) .

5. The point ten=a determines a relative maximum for function f if f is continuous at x=a , and the first derivative f' is positive (+) for ten<a and negative (-) for x>a . The point x=a determines an absolute maximum for office f if information technology corresponds to the largest y-value in the range of f .

6. The point x=a determines a relative minimum for function f if f is continuous at 10=a , and the first derivative f' is negative (-) for x<a and positive (+) for x>a . The point ten=a determines an accented minimum for function f if it corresponds to the smallest y-value in the range of f .

7. The betoken x=a determines an inflection point for function f if f is continuous at x=a , and the second derivative f'' is negative (-) for ten<a and positive (+) for x>a , or if f'' is positive (+) for x<a and negative (-) for 10>a .

8. THE Second DERIVATIVE TEST FOR EXTREMA (This can be used in place of statements five. and 6.) : Assume that y=f(x) is a twice-differentiable function with f'(c)=0 .

a.) If f''(c)<0 and then f has a relative maximum value at ten=c .

b.) If f''(c)>0 and then f has a relative minimum value at x=c .

These are the directions for problems 1 through 10. For each office country the domain. Determine all relative and absolute maximum and minimum values and inflection points. State conspicuously the intervals on which the office is increasing ( $\uparrow$ ) , decreasing ( $\downarrow$ ) , concave up ( $\cup$ ) , and concave downwards ( $\cap$ ) . Make up one's mind 10- and y-intercepts and vertical and horizontal asymptotes when appropriate. Neatly sketch the graph.

Problem 1 : Do detailed graphing for f(x) = x ³ - 3ten ² .

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PROBLEM two : Do detailed graphing for f(ten) = x ⁴ - 410 ⁱⁱⁱ .

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PROBLEM 3 : Do detailed graphing for f(x) = x ³ (ten-2)² .

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Problem 4 : Do detailed graphing for $f(x) = \displaystyle{ 4x \over x^2 + 1 }$ .

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PROBLEM five : Do detailed graphing for $f(x) = \displaystyle{ 2x^2-3x \over x-2 }$ .

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PROBLEM 6 : Do detailed graphing for $f(x) = \displaystyle{ (x-4)^2 \over x^2 - 4 }$ .

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PROBLEM seven : Practice detailed graphing for f(10) = x - 310 ^1/3 .

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PROBLEM 8 : Do detailed graphing for $f(x) = x^{2/3} \Big( \displaystyle{ 5 \over 2 } - x \Big)$ .

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PROBLEM 9 : Do detailed graphing for $f(x) = \sin x - \sqrt{ 3 } \cos x$ for x in $[0, 2\pi ]$ .

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Trouble 10 : Do detailed graphing for $f(x) = x \sqrt{ 4 - x^2 }$ .

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PROBLEM eleven : Consider the cubic polynomial y = A ten ⁱⁱⁱ + vix ² - Bx , where A and B are unknown constants. If possible, determine the values of A and B so that the graph of y has a maximum value at x= -i and an inflection point at x=one .
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