Calculus 1 : How to find rate of change

Study concepts, example questions & explanations for Calculus 1

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Example Questions

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Example Question #801 : How To Find Rate Of Change

The position of a car is defined by the equation .  What is the instantaneous velocity of the car at ?

Possible Answers:

Correct answer:

Explanation:

The instantaneous velocity of the car is the first derivative of the position at a given point.

In this problem,

Example Question #802 : How To Find Rate Of Change

Find  for  .

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we can use either the quotient rule or the product rule.  For this solution, we will use the product rule.  

The product rule states that .  

In this case, let  and .  

Putting both of these together, we get 

.

Example Question #803 : How To Find Rate Of Change

Find the slope of the line tangent to the curve of the multivariable function f(x,y) at

the point

Possible Answers:

-1/2

1

None of the other answers

1/2

-1

Correct answer:

-1

Explanation:

To find the slope of the tangent line at the specified point, we must first verify that the specified point actually exists on the curve. We check that 

Since the verification checks out, the problem has a solution and we can continue with Implicit Differentiation. 

Recall that for Implicit differentiation, if we have a function in terms of y, we have that it's derivative with respect to x is 

Applying this to the given function, we have that 

We must also utilize the Chain Rule to obtain the derivative; we get that

Algebraically, we divide the cosine term to begin isolating dy/dx. We then get that

To obtain the slope of the tangent line, we substitute the specified point (x,y) for x and y respectively. 

Example Question #801 : How To Find Rate Of Change

A balloon's radius is increasing at a rate of 5 cm/s at the exact moment when the radius of the balloon is 1 cm. Assuming that the balloon is a sphere, at what rate is the volume increasing?

Possible Answers:

Correct answer:

Explanation:

The volume of a sphere is 

 

which can also be written as a function with respect to time, i.e.. 

.  

If we take the derivative of this, then 

.  

The problem tells us though that the rate of change of the radius is  and .  

Plugging in these values we find that 

.

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