Question

Calc BC Problem. No random answers plz

Answer 1

Answer:

Part A)

$f(1)=2, \; f^{-1}(1)=0, \; f^\prime(1)=1.4, \; (f^{-1})^\prime(1)=\frac{10}{7}$

Part B)

$y=\frac{5}{14}x+\frac{4}{7}$

Step-by-step explanation:

Please refer to the table of values.

Part A)

A. 1)

We want to find f(1).

According to the table, when x=1, f(x)=2.

Hence, f(1)=2.

A. 2)

We want to find f⁻¹(1).

Notice that when x=0, f(x)=1.

So, f(0)=1.

Then by definition of inverses, f⁻¹(1)=0.

A. 3)

We want to find f’(1).

According to the table, when x=1, f’(x)=1.4.

Hence, f’(1)=1.4.

A. 4)

We will need to do some calculus.

Let g(x) equal to f⁻¹(x). Then by the definition of inverses:

$f(g(x))=x$

Take the derivative of both sides with respect to x. On the left, this will require the chain rule. Therefore:

$f^\prime(g(x))\cdot g^\prime(x)=1$

Solve for g’(x):

$g^\prime(x)=\frac{1}{f^\prime(g(x))}$

Substituting back f⁻¹(x) for g(x) yields:

$(f^{-1})^\prime(x)=\frac{1}{f^\prime({f^{-1}(x)})}$

Therefore:

$(f^{-1})^\prime(1)=\frac{1}{f^\prime({f^{-1}(1)})}$

We already determined previously that f⁻¹(1) is 0. Therefore:

$(f^{-1})^\prime(1)=\frac{1}{f^\prime(0)}$

According to the table, f’(0) is 0.7. So:

$(f^{-1})^\prime(1)=\frac{1}{0.7}=\frac{10}{7}$

Hence, (f⁻¹)’(1)=10/7.

Part B)

We want to find the equation of the tangent line of y=f⁻¹(x) at x=4.

First, let’s determine the points. Since f(2)=4, this means that f⁻¹(4)=2.

Hence, our point is (4, 2).

We will now need to find our slope. This will be the derivative at x=4. Therefore:

$(f^{-1})^\prime(4)=\frac{1}{f^\prime({f^{-1}(4)})}$

We know that f⁻¹(4)=2. So:

$(f^{-1})^\prime(4)=\frac{1}{f^\prime(2)}$

Evaluate:

$(f^{-1})^\prime(4)=\frac{1}{f^\prime(2)}=\frac{1}{2.8}=\frac{10}{28}=\frac{5}{14}$

Now, we can use the point slope form. Our point is (4, 2) and our slope at that point is 5/14.

So:

$y-2=\frac{5}{14}(x-4)$

Solve for y:

$y-2=\frac{5}{14}x-\frac{20}{14}$

Adding 2 to both sides yields:

$y=\frac{5}{14}x-\frac{20}{14}+\frac{28}{14}$

Hence, our equation is:

$y=\frac{5}{14}x+\frac{4}{7}$