Derivatives, Sin and Cos, Rate of Change, Tangent Lines

In summary, the conversation discusses a request for help with calculus problems and provides three questions related to finding the average rate of change and tangent line of a given function. The conversation also emphasizes the importance of making an effort before seeking help.
  • #1
JackieAnne
7
0
Hi, I am in calculus and am having major struggles. If someone could provide a walk through on how to answer these questions, that would be fantastic. Cheers!

Let f(x)=−3x+6 if x<-3
= 15 if x > -3

Find the average rate of change of f(x) on the interval −5<x<5 .

The average rate of change of f(x) on the interval −5<x<5 is ?


Consider the function f(x)=−7/x+4.

We will take steps to find the tangent line to the graph of f at the point (−7,−3/−7).

(a) Let (xf(x)) be a point on the graph of f with x=−7 . The slope of the (secant) line joining the two points (−7,−3/−7) and (xf(x)) can be simplified to the form A/x+4, where A is a constant. Find A.

Answer: A= .

(b) By considering the slope of the secant line as x approaches −7, find the slope of the tangent line to the graph of f at the point (−7,−3/−7).

Answer: The slope of the tangent line to the graph of f at the point (−7,−3/−7) is .

(c) Find the equation of the tangent line to the graph of f at the point (−7,−3/−7). Write your answer in the form y=mx+b.
 
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  • #2
You have to provide us with a step by step attempt at a solution, so we can try to help you where you're struggling. Nobody is going to solve your homework, this is not the point of this forum. We're homework helpers not solvers.
 
  • #3
JackieAnne said:
Hi, I am in calculus and am having major struggles. If someone could provide a walk through on how to answer these questions, that would be fantastic. Cheers!

Let f(x)=−3x+6 if x<-3
= 15 if x > -3

Find the average rate of change of f(x) on the interval −5<x<5 .

The average rate of change of f(x) on the interval −5<x<5 is ?


Consider the function f(x)=−7/x+4.

We will take steps to find the tangent line to the graph of f at the point (−7,−3/−7).

(a) Let (xf(x)) be a point on the graph of f with x=−7 . The slope of the (secant) line joining the two points (−7,−3/−7) and (xf(x)) can be simplified to the form A/x+4, where A is a constant. Find A.

Answer: A= .

(b) By considering the slope of the secant line as x approaches −7, find the slope of the tangent line to the graph of f at the point (−7,−3/−7).

Answer: The slope of the tangent line to the graph of f at the point (−7,−3/−7) is .

(c) Find the equation of the tangent line to the graph of f at the point (−7,−3/−7). Write your answer in the form y=mx+b.

What have you tried? Before we can offer help, you must have made an effort on your own behalf.
 

1. What are derivatives and why are they important?

Derivatives are mathematical tools used to measure the rate of change of a function at a specific point. They are important because they allow us to understand the behavior and properties of functions, which has applications in various fields such as physics, economics, and engineering.

2. How are derivatives related to sin and cos?

Derivatives of the sine and cosine functions are used to calculate the instantaneous rate of change of these trigonometric functions. This is particularly useful in physics and engineering, where these functions are commonly used to model periodic phenomena.

3. What is the rate of change and how is it calculated?

The rate of change is a measure of how much a quantity changes with respect to another quantity. It is calculated by finding the slope of the tangent line to a curve at a specific point. This can be done using the derivative of the function at that point.

4. How are tangent lines related to derivatives?

Tangent lines are lines that touch a curve at a specific point and have the same slope as the curve at that point. They are related to derivatives because the slope of a tangent line can be calculated using the derivative of the function at that point.

5. How are derivatives, sin and cos, rate of change, and tangent lines used in real life?

These concepts are used in various real-life applications, including physics, engineering, economics, and finance. For example, derivatives are used in physics to calculate the velocity and acceleration of an object, while in finance, they are used to calculate the risk and return of financial assets. Tangent lines are used in engineering to determine the slope and direction of a road or pipeline, and in economics, they are used to calculate marginal cost and revenue.

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