How to find the equation of the line tangent to the graph of f at the indicated value of x? f(x)= 4 - lnx ; x=1

Answer 1

#x+y=5#

#"we will use the equation: "y-y_1=m(x-x_1)#
#"where "(x_1,y_1)" is a known point on the line and " m= "the gradient of the line at that point"#
#" to find the gradient we need to calculate "f'(x_1)#
#f(x)=4-lnx#
#y_1=f(x_1)=f(1)=4-ln1=4-0=4#
#color(blue)((x_1,y_1)=(1,4))#
#f(x)=4-lnx=>f'(x)=-1/x#
#color(red)(m=f'(1)=-1/1=-1)#
#y-color(blue)(y_1)=color(red)(m)(x-color(blue)(x_1))#
#y-4=-1(x-1)#
#y-4=-x+1#
#x+y=5#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the equation of the line tangent to the graph of f at x=1, we need to find the derivative of f(x) and evaluate it at x=1.

The derivative of f(x) can be found using the power rule and the chain rule. The derivative of 4 is 0, and the derivative of ln(x) is 1/x. Therefore, the derivative of f(x) is 0 - 1/x.

Evaluating the derivative at x=1, we have 0 - 1/1 = -1.

So, the slope of the tangent line at x=1 is -1.

To find the equation of the line, we need a point on the line. Since the line is tangent to the graph of f at x=1, we can use the point (1, f(1)).

Plugging x=1 into f(x), we have f(1) = 4 - ln(1) = 4 - 0 = 4.

Therefore, the point on the line is (1, 4).

Using the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, we can substitute the values:

y - 4 = -1(x - 1)

Simplifying, we have y - 4 = -x + 1.

Rearranging the equation, we get y = -x + 5.

So, the equation of the line tangent to the graph of f at x=1 is y = -x + 5.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7