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Evaluate the limit of the given expression and find the first and second derivatives of the given polynomial.

Mathematics

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2 parts
i

40x4+9x2+8x40x^4 + 9x^2 + 8x

ii

160x3+18x+8160x^3 + 18x + 8

Step-by-step solution

4 steps

9. (a) Evaluate lim⁡x→1(x2−7x+10x2−4)\lim_{x \to 1} \left(\frac{x^2 - 7x + 10}{x^2 - 4}\right)

Step 1: Substitute x=1x=1 into the expression.

lim⁡x→1(x2−7x+10x2−4)=(1)2−7(1)+10(1)2−4\lim_{x \to 1} \left(\frac{x^2 - 7x + 10}{x^2 - 4}\right) = \frac{(1)^2 - 7(1) + 10}{(1)^2 - 4}

Step 2: Simplify the expression.

=1−7+101−4=4−3= \frac{1 - 7 + 10}{1 - 4} = \frac{4}{-3} −43\boxed{-\frac{4}{3}}

9. (b) If y=8x5+3x3+4x2y = 8x^5 + 3x^3 + 4x^2 find

Part i· 2 steps

dydx\frac{dy}{dx}

  1. Differentiate each term of yy with respect to xx using the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}.

    dydx=ddx(8x5)+ddx(3x3)+ddx(4x2)\frac{dy}{dx} = \frac{d}{dx}(8x^5) + \frac{d}{dx}(3x^3) + \frac{d}{dx}(4x^2) dydx=8(5)x5−1+3(3)x3−1+4(2)x2−1\frac{dy}{dx} = 8(5)x^{5-1} + 3(3)x^{3-1} + 4(2)x^{2-1}

  2. Simplify the expression.

    40x4+9x2+8x\boxed{40x^4 + 9x^2 + 8x}

Part ii

160x3+18x+8160x^3 + 18x + 8

d2ydx2\frac{d^2y}{dx^2}

  1. Differentiate the expression for dydx\frac{dy}{dx} with respect to xx.

    d2ydx2=ddx(40x4+9x2+8x)\frac{d^2y}{dx^2} = \frac{d}{dx}(40x^4 + 9x^2 + 8x) d2ydx2=ddx(40x4)+ddx(9x2)+ddx(8x)\frac{d^2y}{dx^2} = \frac{d}{dx}(40x^4) + \frac{d}{dx}(9x^2) + \frac{d}{dx}(8x) d2ydx2=40(4)x4−1+9(2)x2−1+8(1)x1−1\frac{d^2y}{dx^2} = 40(4)x^{4-1} + 9(2)x^{2-1} + 8(1)x^{1-1}

  2. Simplify the expression.

    d2ydx2=160x3+18x1+8x0\frac{d^2y}{dx^2} = 160x^3 + 18x^1 + 8x^0 160x3+18x+8\boxed{160x^3 + 18x + 8}
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