1. Determine the average rate of change. 2. Find an equation for the line that is tangent to the graph of the function at x = 1. 3. Calculate the slope of the tangent to the given... 1. Determine the average rate of change. 2. Find an equation for the line that is tangent to the graph of the function at x = 1. 3. Calculate the slope of the tangent to the given function at the given point value of x. a. f(x) = 3/(x+1), P(2, 1) b. g(x) = sqrt(x + 2), P(-1, 1) c. h(x) = 2/sqrt(x+5), P(4, 2/3) d. f(x) = 5/(x-2), P(4, 5/2) 4. Calculate the slope of the graph of f(x) = {4 - x^2, if x <= 1; 2x + 1, if x > 1} at each of the following points: a. P(-1,3) b. P(2,5) 5. The height, in metres, of an object that has fallen from a height of 180 Read more

Steps to Solve

1. Problem 2a: Find the derivative of $f(x) = \frac{3}{x+1}$

To find the slope of the tangent line, we need to find the derivative of the function $f(x)$. We can rewrite the function as $f(x) = 3(x+1)^{-1}$. Using the power rule and chain rule, we have: $f'(x) = -3(x+1)^{-2} = \frac{-3}{(x+1)^2}$

2. Problem 2a: Evaluate the derivative at $x=2$

Now, we need to evaluate the derivative at $x=2$ to find the slope of the tangent at the point $P(2,1)$: $f'(2) = \frac{-3}{(2+1)^2} = \frac{-3}{3^2} = \frac{-3}{9} = -\frac{1}{3}$

3. Problem 2b: Find the derivative of $g(x) = \sqrt{x+2}$

We can rewrite the function as $g(x) = (x+2)^{\frac{1}{2}}$. Using the power rule and chain rule, we have: $g'(x) = \frac{1}{2}(x+2)^{-\frac{1}{2}} = \frac{1}{2\sqrt{x+2}}$

4. Problem 2b: Evaluate the derivative at $x=-1$

Now, we need to evaluate the derivative at $x=-1$ to find the slope of the tangent at the point $P(-1,1)$: $g'(-1) = \frac{1}{2\sqrt{-1+2}} = \frac{1}{2\sqrt{1}} = \frac{1}{2}$

5. Problem 2c: Find the derivative of $h(x) = \frac{2}{\sqrt{x}+5}$

Rewrite the function as $h(x) = \frac{2}{x^{1/2} + 5}$. Then, the derivative is: $h'(x) = \frac{-2(\frac{1}{2}x^{-1/2})}{(x^{1/2} + 5)^2} = \frac{-x^{-1/2}}{(x^{1/2} + 5)^2} = \frac{-1}{\sqrt{x}(\sqrt{x}+5)^2}$

6. Problem 2c: Evaluate the derivative at $x=4$

Now, we need to evaluate the derivative at $x=4$: $h'(4) = \frac{-1}{\sqrt{4}(\sqrt{4}+5)^2} = \frac{-1}{2(2+5)^2} = \frac{-1}{2(7)^2} = \frac{-1}{2(49)} = -\frac{1}{98}$

7. Problem 2d: Find the derivative of $f(x) = \frac{5}{x-2}$

Rewrite the function as $f(x) = 5(x-2)^{-1}$. Using the power rule and chain rule, we have: $f'(x) = -5(x-2)^{-2} = \frac{-5}{(x-2)^2}$

8. Problem 2d: Evaluate the derivative at $x=4$

Now, we need to evaluate the derivative at $x=4$: $f'(4) = \frac{-5}{(4-2)^2} = \frac{-5}{2^2} = \frac{-5}{4}$

9. Problem 3a: Find the slope of $f(x)$ at $P(-1,3)$

$f(x) = \begin{cases} 4-x^2, & \text{if } x \leq 1 \ 2x+1, & \text{if } x > 1 \end{cases}$

Since $x = -1 \le 1$, we use $f(x) = 4 - x^2$.

$f'(x) = -2x$ $f'(-1) = -2(-1) = 2$

10. Problem 3b: Find the slope of $f(x)$ at $P(2,5)$

Since $x=2 > 1$, we use $f(x) = 2x+1$.

$f'(x) = 2$ $f'(2) = 2$