Question

(this problem is similar to the example in your textbook about guessing a formula for the derivative of f(x) = x² and to a similar problem in the exercises. study that example carefully before doing this problem.) let f(t)=1t² + 6t. estimate the following to within two decimal places by using small enough intervals. a. f(5)≈ b. f(-3)≈ c. f(10)≈ 3. determine a formula for f(t).

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For the function $f(t)=1t^{2}+6t$, we can find its derivative term - by - term.

Step2: Differentiate each term

For the first term $t^{2}$, using the power - rule with $a = 1$ and $n = 2$, the derivative is $2t^{2-1}=2t$. For the second term $6t$, using the power - rule with $a = 6$ and $n = 1$, the derivative is $6\times1\times t^{1 - 1}=6$. So, $f^\prime(t)=2t + 6$.

Step3: Find $f^\prime(5)$

Substitute $t = 5$ into $f^\prime(t)$. Then $f^\prime(5)=2\times5+6=10 + 6=16$.

Step4: Find $f^\prime(-3)$

Substitute $t=-3$ into $f^\prime(t)$. Then $f^\prime(-3)=2\times(-3)+6=-6 + 6=0$.

Step5: Find $f^\prime(10)$

Substitute $t = 10$ into $f^\prime(t)$. Then $f^\prime(10)=2\times10+6=20 + 6=26$.

Answer:

a. $f^\prime(5)=16$
b. $f^\prime(-3)=0$
c. $f^\prime(10)=26$
d. $f^\prime(t)=2t + 6$