Calculus - Tangent Line Question

• rakalakalili
In summary, the problem is to find the value of b that makes the line y=10x tangent to the curve y=e^{bx} at some point in the xy-plane. This can be solved by setting up a system of equations and using the fact that the derivative of e^{bx} is equal to the slope of the tangent line. After some algebraic manipulation, the solution is x = e/10.
rakalakalili

Homework Statement

Hello, this is a problem from the practice test for the GRE subject test.
For what value of b is the line $$y=10x$$ tangent to the curve $$y=e^{bx}$$ at some point in the xy-plane?
A) $$\frac{10}{e}$$
B)10
C)10e
D)$$e^{10}$$
E)e

The Attempt at a Solution

For the line to be tangent to the curve, they must intersect at a point and their derivatives must be equal. So we get a system of equations:
$$10x=e^{bx}$$
$$10=be^{bx}$$
But I cannot think of a way to solve this system easily. Would the best method be to simply plug in the possible answers and check? This seems time consuming, and since this is a practice GRE question I am interested in the simplest way of solving it. Please let me know if I am over looking something, thank you!

Hint:
e^bx= 1+bx+o(x^2)
and we have that (e^bx-1)/x=b+o(x)

So because the derivative of e^bx should be equal the slope of the tangent line, we concur?

I just picked the point zero randomly.

You have to solve two equations for x and for b:
10x = e^(bx) and 10 = b*e^(bx)

Multiplying the second eqn by x and comparing it to the first eqn gives

e^bx = 10x = x*b*e^(bx)

It follows that 1 = x*b, so x = 1/b. A little more work will reveal that x = e/10.

1. What is the purpose of finding a tangent line using calculus?

The purpose of finding a tangent line using calculus is to determine the rate of change of a function at a specific point. This can be useful in various real-world applications, such as determining the velocity of an object at a certain time or the growth rate of a population.

2. How do you find the equation of a tangent line using calculus?

To find the equation of a tangent line using calculus, you first need to find the derivative of the function at the given point. This derivative represents the slope of the tangent line. Then, using the point-slope formula, you can plug in the slope and the coordinates of the given point to find the equation of the tangent line.

3. Can you only find the tangent line using calculus for linear functions?

No, you can use calculus to find the tangent line for any type of function, including nonlinear functions. The key is to find the derivative of the function at the given point, which represents the slope of the tangent line.

4. How does finding the tangent line relate to the concept of limits in calculus?

The concept of limits is essential in finding the tangent line using calculus. When finding the derivative, you are essentially finding the limit of the slope of the secant line as the distance between two points approaches zero. This limit represents the slope of the tangent line.

5. Are there any real-world applications of finding the tangent line using calculus?

Yes, there are many real-world applications of finding the tangent line using calculus. Some examples include determining the optimal angle for a ramp or roller coaster, calculating the rate of change of a stock price, and finding the maximum or minimum value of a function.

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