Math Problem Statement

Find an equation of the tangent line to the curve at the given point: y = (1 + 4x)^9, (0, 1).

Solution

We are tasked with finding the equation of the tangent line to the curve y=(1+4x)9y = (1 + 4x)^9 at the point (0,1)(0, 1).

Steps to solve:

  1. Find the derivative of the curve:
    The slope of the tangent line is given by the derivative of the function at the point. So, we need to compute the derivative dydx\frac{dy}{dx}.

    y=(1+4x)9y = (1 + 4x)^9 Using the chain rule, the derivative of yy with respect to xx is: dydx=9(1+4x)8ddx(1+4x)\frac{dy}{dx} = 9(1 + 4x)^8 \cdot \frac{d}{dx}(1 + 4x) The derivative of 1+4x1 + 4x is 4, so: dydx=9(1+4x)84\frac{dy}{dx} = 9(1 + 4x)^8 \cdot 4 dydx=36(1+4x)8\frac{dy}{dx} = 36(1 + 4x)^8

  2. Evaluate the derivative at the point (0,1)(0, 1):
    Now, substitute x=0x = 0 into the derivative: dydxx=0=36(1+4(0))8=36×18=36\frac{dy}{dx} \bigg|_{x = 0} = 36(1 + 4(0))^8 = 36 \times 1^8 = 36 So, the slope of the tangent line at (0,1)(0, 1) is 36.

  3. Use the point-slope form of the equation of a line:
    The point-slope form of a line is: yy1=m(xx1)y - y_1 = m(x - x_1) where mm is the slope and (x1,y1)(x_1, y_1) is the point on the line. Here, m=36m = 36, x1=0x_1 = 0, and y1=1y_1 = 1. Thus, the equation becomes: y1=36(x0)y - 1 = 36(x - 0) Simplifying: y1=36xy - 1 = 36x y=36x+1y = 36x + 1

Final Answer:

The equation of the tangent line to the curve at the point (0,1)(0, 1) is: y=36x+1y = 36x + 1

Would you like further details or clarifications?

Here are five related questions to explore:

  1. What happens to the tangent line equation if the point is shifted to a different point on the curve?
  2. How does the chain rule apply to functions raised to high powers, like in this problem?
  3. What role does the point-slope form play in finding the tangent line?
  4. How can we use the derivative to analyze the behavior of the curve near x=0x = 0?
  5. What is the geometrical interpretation of the derivative at x=0x = 0?

Tip: The chain rule is especially useful for functions that involve compositions, such as powers of expressions. Make sure to differentiate both the outer and inner functions properly.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Tangent Lines
Chain Rule

Formulas

Derivative of y = (1 + 4x)^9: dy/dx = 36(1 + 4x)^8
Point-slope form of a line: y - y1 = m(x - x1)

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12