At what point on the curve is there a tangent line parallel to the line

Click For Summary
SUMMARY

The tangent line to the curve defined by the function \(y = e^x\) is parallel to the line \(y = 2x\) when the slope of the curve equals the slope of the line. The derivative of \(y\) is \(\frac{dy}{dx} = e^x\). To find the point of tangency, set \(e^x = 2\) and solve for \(x\). This results in \(x = \ln(2)\), confirming that the tangent line at this point is parallel to the line \(y = 2x\).

PREREQUISITES
  • Understanding of derivatives and slopes in calculus
  • Familiarity with exponential functions, specifically \(e^x\)
  • Knowledge of logarithmic functions, particularly natural logarithms
  • Graphing skills to visualize functions and their tangents
NEXT STEPS
  • Study the properties of exponential functions and their derivatives
  • Learn how to solve equations involving natural logarithms
  • Explore the concept of tangent lines in calculus
  • Practice graphing exponential functions and their tangents
USEFUL FOR

Students and educators in calculus, mathematicians interested in function analysis, and anyone looking to deepen their understanding of derivatives and tangent lines.

tmt1
Messages
230
Reaction score
0
At what point on the curve
$$y = e^x$$ is the tangent line parallel to the line

$$y = 2x$$

The derivative of y is

$$\frac{dy}{dx} = e^x$$

But I'm unsure how to proceed from here.
 
Physics news on Phys.org
tmt said:
At what point on the curve
$$y = e^x$$ is the tangent line parallel to the line

$$y = 2x$$

The derivative of y is

$$\frac{dy}{dx} = e^x$$

But I'm unsure how to proceed from here.

You need to equate the slope of the exponential function at a point $x$, which is $e^x$ (as you have found) with the slope of the line $y = 2x$, which is $2$. Hence $e^x = 2$, and solve for $x$. Then if you graph that, you will find that the tangent line to $e^x$ at that point $x$ is parallel to $y = 2x$ as they have the same slope :)
 

Similar threads

High School How to Find the Intersection of a Logarithmic Curve and a Tangent Line?
  • · Replies 14 ·
Replies
14
Views
2K
MHB How Can I Find the Equation of the Dotted Tangent Line of a Circle?
  • · Replies 5 ·
Replies
5
Views
2K
High School Tangents to a curve as functions of different variables
  • · Replies 5 ·
Replies
5
Views
2K
MHB Equation for tangent of the curve
  • · Replies 1 ·
Replies
1
Views
1K
MHB 205.2.6.1-2 another implicit problem with tangent line.
  • · Replies 1 ·
Replies
1
Views
2K
MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines
  • · Replies 3 ·
Replies
3
Views
2K
MHB 210 AP Calculus Exam problem tangent line to curve
  • · Replies 2 ·
Replies
2
Views
2K
MHB How Does the Derivative of a Cubic Function Result in a Tangent Line?
  • · Replies 4 ·
Replies
4
Views
3K
Precise definition of tangent line to a curve
  • · Replies 18 ·
Replies
18
Views
3K
Undergrad Calc 1 practice exam question -- finding the y intercept of the tangent line
  • · Replies 6 ·
Replies
6
Views
2K