I know it's hard to draw a graph here, but I just need an explanation of both if possible. I appreciate any help I can get! :)
"Determine whether the following statements are true and give an explanation or counterexample. Explain why or why not, using complete sentences and appropriate terminology correctly. For part a, include a nice graph to support your claim.
a. Consider the graph of the parabola f(x)=x^2. For a>0 and h>0, the secant line through (a,f(a)) and (a+h,f(a+h)) and always has greater slope than the tangent line at (a+f(a)).
b. d/dx (π^5)=5π^4 "
The secant line has slope
((a+h)^2 - a^2)/h = (a^2+2ah+h^2-a^2)/h = (2a+h)
The slope of the tangent line is 2a
I'm sure you can graph y=x^2, right?
d/dx π^5 = 0
because π is a constant
However,
d/dx x^5 = 5x^4
so at x=π, the tangent line has slope 5π^4
Thanks for your help! I mostly struggled with the explanations, but the graphing was fine.
a. To determine whether the secant line always has a greater slope than the tangent line, we need to compare their slopes.
First, let's find the slope of the secant line. The secant line passing through points (a, f(a)) and (a+h, f(a+h)) is given by the formula:
slope_secant = (f(a+h) - f(a)) / (a+h - a)
Simplifying this expression:
slope_secant = (f(a+h) - f(a)) / h
Then, let's find the slope of the tangent line. The slope of the tangent line at a point (a, f(a)) on the curve y = f(x) can be found by taking the derivative of f(x) with respect to x and evaluating it at x=a.
Differentiating f(x) = x^2:
f'(x) = 2x
Evaluating at x=a:
f'(a) = 2a
So the slope of the tangent line is 2a.
Now, let's examine if the slope_secant is always greater than slope_tangent, i.e., if (f(a+h) - f(a)) / h > 2a.
To provide a counterexample, we need to find values of a and h for which the inequality does not hold. Let's consider a = 1 and h = 0.5 (you can choose other values to verify this as well).
Using these values:
slope_secant = (f(1+0.5) - f(1)) / 0.5
= (f(1.5) - 1) / 0.5
= (1.5^2 - 1) / 0.5
= (2.25 - 1) / 0.5
= 1.25 / 0.5
= 2.5
slope_tangent = 2a
= 2(1)
= 2
Comparing the two slopes, we find that slope_tangent (2) is greater than slope_secant (2.5). Therefore, the claim is false.
Here is an additional nice graph to visually support the counterexample:
```
^
|
3 |
| o (1,1)
2 | o
| o (1.5,2.25)
1 | o /
| * (1,2)
0 |_______o_______
| 0 1 2 --> x-axis
```
b. To determine whether the derivative of π^5 is 5π^4, we can use the power rule of differentiation.
The power rule states that if f(x) = x^n, then d/dx (f(x)) = n*x^(n-1).
Now, let's apply the power rule to f(x) = π^5:
d/dx (π^5) = 5*π^4
Therefore, the statement is true, and the derivative of π^5 is indeed 5π^4.