Question

Consider the function f(x) = x2 - 8x + 8. The slope of the tangent line at x = 5 is 2. Find the equation of this tangent line. (Write your answer in the form y=mx+b with no spaces - as always, if the computer marks you incorrect even if you have the answer correct but just in a different form than I use, email me and I will fix your points.) Given the function f(x)= 4x2 - 3x + 1, calculate the derivative of this function. (Write your answer as a polynomial from highest to lowest degree. For example: x^3-5x^2-6x+7) The height of a ball thrown into the air is given by the function h(t)= 32t - 16t2. If you were to calculate the derivative, you would get h' (t) = 32 - 32t. Find the instantaneous velocity at t = 0.1 sec. Write your answer without any units.

Answer 1

(a)

f(1) = x2 – 81 +8.

$\implies f(5)=5^2-8*5+8.=25-40+8=-7$

Equation of the tangent at $x=5$ is given by:

y-f(5) = f'(5) (x - 5)

y+7= 2 (- 5)

$\implies y=2x-10-7$

$\therefore y=2x-17$ is the equation of the tangent at $x=5$ .

(b)

$f(x)=4x^2-3x+1$

$\implies f'(x)=4\times \frac{d}{dx}\, x^2-3\times \frac{d}{dx}\, x+ \frac{d}{dx}\, 1$

$=4\times 2x-3\times 1+ 0$

$=8x-3$

(c)

$h(t)=32t-16t^2$

$\implies h'(t)=32-32t$

Assuming the ball is thrown in a vertical fashion, the instantaneous velocity is at $t=0.1$ is given by

$h'(0.1)=32-32*0.1=32-3.2=28.8$