-apc.4.1.2 Find the slope at x=4

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In summary: because in the equation of the tangent line it saysf(4)= -4^2+4\sqrt{4}= -16+ 8= -8 which is the same as the equation of the graph y=-7x+ 20
  • #1
karush
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$\tiny{apc.4.1.2}$
Find the slope of the tangent line to the graph of
$f(x)=-x^2+4\sqrt{x}$ at $x=4$
$a.\ 8\quad b.\ -10\quad c.\ -9\quad d.\ -5\quad e.\ -7$

$f'(x)=-2x+\frac{2}{\sqrt{x}}$
$m=f'(4)=-2(4)+\frac{2}{\sqrt{4}}=-7$ which is (e)

however not asked for here but I forgot how to find $b$ of $y=-7x+b$
 
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  • #2
$y - f(4) = -7(x-4)$

clean it up ...
 
  • #3
skeeter said:
$y - f(4) = -7(x-4)$

clean it up ...

that didn't seem to be a tangent line to f{x}
$-8=-7(4)+b$
$-8+28=20=b$

$y=-7x+20$
 
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  • #4
the equation in my previous post is in point-slope form, $y - y_1 = m(x - x_1)$ where $m = -7, \, x_1 = 4 \text{ and } y_1 = f(4)$

$f(4) = -8$

$y + 8 = -7(x - 4) \implies y = -7x + 20$
 
  • #5
karush said:
that didn't seem to be a tangent line to f{x}
$-8=-7(4)+b$
$-8+28=20=b$

$y=-7x+20$
Yes, it is. Since $f(x)= -x^2+ 4\sqrt{x}$ so $f(4)= -4^2+4\sqrt{4}= -16+ 8= -8$. That is, the expression skeeter gave, $y- f(4)= -7(x- 4)$ becomes $y- (-8)= y+ 8= -7x+ 28$ so, subtracting 8 from both sides, $y= -7x+ 20$ as you have.
 
  • #6
Is the option wrong? I used cameramath and the answer is -8

20201210-0.jpg
 
  • #7
TiffanyBK said:
Is the option wrong? I used cameramath and the answer is -8

View attachment 10874
The answer to what question? You want to find the equation of the tangent line to $f(x)= -x^2+ 4\sqrt{x}$ at x= 4. You have already determined that the derivative of f, so the slope of the tangent line, at x= 4 is -7. Thus you know that the equation of the tangent line is y= -7x+ b for some number, b. So what is the value of y at x= 4? $y= f(4)= -(4^2)+ 4\sqrt{4}= -16+ 4(2)= -8$ -8 is the value of f(4). But that is not necessarily b! Now we know that we must have y= -7(4)+ b= -28+ b= -8. Adding 28 to both sides, b= 20.

The equation of the tangent line at x= 4 is y= -7x+ 20.
 
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  • #8
well i think the confusion was f(4) vs f'(4)
 

Related to -apc.4.1.2 Find the slope at x=4

What is the meaning of "-apc.4.1.2 Find the slope at x=4"?

The "-apc.4.1.2" refers to a specific mathematical function or equation. "Find the slope" indicates that we are looking for the rate of change of this function. "x=4" specifies the point at which we want to find the slope.

Why is finding the slope important?

Finding the slope of a function at a specific point can help us understand the behavior of the function and make predictions about its future values. It is also a fundamental concept in calculus and is used in many real-world applications such as physics, engineering, and economics.

What is the formula for finding the slope at a specific point?

The formula for finding the slope at a specific point on a function is given by the derivative of the function at that point. In this case, we would use the formula: slope = f'(4), where f'(x) represents the derivative of the function f(x).

How do you find the derivative of a function?

The derivative of a function is found by taking the limit of the slope of a secant line as the two points on the function get closer and closer together. In other words, it is the instantaneous rate of change of the function at a specific point. There are various methods for finding derivatives, such as using the power rule, product rule, or chain rule.

What are some real-world applications of finding the slope at a specific point?

Finding the slope at a specific point can be used to calculate velocity, acceleration, and other rates of change in physics and engineering. It can also be used in economics to analyze supply and demand curves and determine optimal production levels. In business, finding the slope can help with forecasting and decision making. Additionally, it is used in data analysis to identify trends and patterns in data.

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