Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

For the function
f(x)=-7x^(2)+3x-5, find the slope of the tangent line at
x=3.
Answer:

For the function $f(x)=-7 x^{2}+3 x-5$ , find the slope of the tangent line at $x=3$ . $\newline$ Answer:

View step-by-step help

View step-by-step help

Find the slope of a tangent line using limits

Full solution

Q. For the function

f(x)=-7 x^{2}+3 x-5

, find the slope of the tangent line at

x=3

.

\newline

Answer:

Find Derivative: To find the slope of the tangent line to the function at a specific point, we need to find the derivative of the function, which gives us the slope of the tangent line at any point $x$ . The function is $f(x) = -7x^2 + 3x - 5$ . We will use the power rule for differentiation, which states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Apply Power Rule: Differentiate the function with respect to $x$ . The derivative of $-7x^2$ is $-14x$ (using the power rule, bringing down the exponent $2$ and subtracting $1$ from the exponent). The derivative of $3x$ is $3$ (since the derivative of $x$ is $1$ ). The derivative of a constant, $-5$ , is $-7x^2$ $0$ (since constants do not change and their derivative is always $-7x^2$ $0$ ). So, the derivative $-7x^2$ $2$ is $-7x^2$ $3$ .
Evaluate at $x=3$ : Now we need to evaluate the derivative at $x = 3$ to find the slope of the tangent line at that point. $\newline$ Substitute $x = 3$ into the derivative $f'(x) = -14x + 3$ . $\newline$ $f'(3) = -14(3) + 3 = -42 + 3 = -39$ .
Calculate Slope: The slope of the tangent line at $x = 3$ is $-39$ .

More problems from Find the slope of a tangent line using limits

Question

Consider the curve given by the equation

x y^{2}+5 x y=50

. It can be shown that

\frac{d y}{d x}=\frac{-y(y+5)}{x(2 y+5)} \text {. }

\newline

Write the equation of the vertical line that is tangent to the curve.

Posted 7 months ago

Question

What is the value of

\frac{d}{d x}\left(x^{-3}\right)

x=3

Posted 2 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of algae in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)

\newline

t=0

, the number of algae in the tank is

174

and is increasing at a rate of

28

algae per minute. At what value of

P

P(t)

growing the fastest?

\newline

Posted 9 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of bacteria in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2}{9849} P(598-P)

\newline

t=0

, the number of bacteria in the tank is

196

and is increasing at a rate of

16

bacteria per minute. At what value of

P

does the graph of

P(t)

have an inflection point?

\newline

Posted 9 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{45}{125404} P(800-P)

\newline

t=0

, the number of people infected by the disease is

214

and is increasing at a rate of

45

people per hour. What is the limiting value for the total number of people infected by the disease as time increases?

\newline

Posted 9 months ago

Question

The exponential function

f

is graphed in the

x y

x

1, y

increases by a factor of

3

. Which of the following could be

f

\newline

1

\newline

f(x)=\left(\frac{1}{3}\right)^{x}

\newline

f(x)=\left(\frac{1}{3}\right)^{x}+3

\newline

f(x)=3^{x}+2

\newline

f(x)=2(3)^{x}

Posted 2 months ago

Question

Which of the following is a correct interpretation of the expression

\newline

-4+13

\newline

1

\newline

(A) The number that is

4

-13

on the number line

\newline

(B) The number that is

4

to the right of

-13

on the number line

\newline

(C) The number that is

13

-4

on the number line

\newline

(D) The number that is

13

to the right of

-4

on the number line

Posted 8 months ago

Question

f

f(x)=x^{3}+2 \sin (3 x-3)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=0.5

. You should round all decimals to

3

\newline

Posted 9 months ago

Question

f

f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=-2.5

. You should round all decimals to

3

\newline

Posted 9 months ago

Question

f

f(x)=x^{2}+x-2 \sin (2 x)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=3

. You should round all decimals to

3

\newline

Posted 9 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant