Question
If $f(x) = x^3 + 7x^2 + 8x - 9,$ find f(4).
✓
Answer
$f(x) = x^3 + 7x^2 + 8x - 9$ is a polynomial function.
So, it is differentiable every.
$\text{f}'(4)=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(4+\text{h})-\text{h}(4)}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[(4+\text{h})^3+7(4+\text{h})^2+8(4+\text{h})-9]-[64+112+32-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[64+\text{h}^3+48\text{h}+12\text{h}^2+112+7\text{h}^2+56\text{h}+32+8\text{h}-9]-[210-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}+210-9-210+9}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}(\text{h}^2+19\text{h}+112)}{\text{h}}$
$\text{f}'(4)=112$
So, it is differentiable every.
$\text{f}'(4)=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(4+\text{h})-\text{h}(4)}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[(4+\text{h})^3+7(4+\text{h})^2+8(4+\text{h})-9]-[64+112+32-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{[64+\text{h}^3+48\text{h}+12\text{h}^2+112+7\text{h}^2+56\text{h}+32+8\text{h}-9]-[210-9]}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}+210-9-210+9}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}^3+19\text{h}^2+112\text{h}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}(\text{h}^2+19\text{h}+112)}{\text{h}}$
$\text{f}'(4)=112$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Solve the following determinant equations:
$\begin{vmatrix}1&1&\text{x}\\\text{p}+1&\text{p}+1&\text{p}+\text{x}\\3&\text{x}+1&\text{x}+2\end{vmatrix}=0$
→$\begin{vmatrix}1&1&\text{x}\\\text{p}+1&\text{p}+1&\text{p}+\text{x}\\3&\text{x}+1&\text{x}+2\end{vmatrix}=0$
Find the foot of the perpendicular from (0, 2, 7) on the line $\frac{\text{x}+2}{-1}=\frac{\text{y}-1}{3}=\frac{\text{z}-3}{-2}.$
→Find the area of the region $\Bigg\{(\text{x},\text{y}): \frac{\text{x}^{2}}{\text{a}^{2}}+\frac{\text{y}^{2}}{\text{b}^{2}}<1< \frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}\bigg\}$
→A manufacturer produces three types of bolts, $x, y$ and $z$ which he sells in two markets. Annual sales (in ₹) are indicated below:
If unit sales prices of $x, y$ and z are ₹ 2.50, ₹ $1.50$ and ₹ $1.00$ respectively, then answer the following questions using the concept of matrices.
→
| Markets | Products | ||
| $X$ | $Y$ | $Z$ | |
| I | $10000$ | $2000$ | $18000$ |
| II | $6000$ | $20000$ | $8000$ |
- Find the total revenue collected from the Market-I.
- $₹\ 44000$
- $₹\ 48000$
- $₹\ 46000$
- $₹\ 53000$
- Find the total revenue collected from the Market-II.
- $₹\ 51000$
- $₹\ 53000$
- $₹\ 46000$
- $₹\ 49000$
- If the unit costs of the above three commodities are $₹\ 2.00, ₹\ 1.00$ and $50$ paise respectively, then find the gross profit from both the markets.
- $₹\ 53000$
- $₹\ 46000$
- $₹\ 34000$
- $₹\ 32000$
- If matrix $\text{A}=[\text{a}_\text{ij}]_{2\times2},$ where $\text{a}_\text{ij}=1,$ if $\text{i}\neq\text{j}$ and $\text{a}_\text{ij}=0,$ if $\text{i}=\text{j}$ then $A^2$ is equal to:
- $I$
- $A$
- $OR$
- None of these
- If $A$ and $B$ are matrices of same order, then $(AB' - BA')$ is a.
- Skew-synunetric matrix.
- Null matrix.
- Symmetric matrix.
- Unit matrix.
A factory has three machines X, Y and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?
Prove that $\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{x}<0\\\text{x}+1,&\text{x}\geq0\end{cases}$ is everywhere continuous.
→Show that the points $\text{A}\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big),\ \text{B}\big(\hat{\text{i}}-3\hat{\text{j}}-5\hat{\text{k}}\big),$ $\text{C}\big(3\hat{\text{i}}-4\hat{\text{j}}-4\hat{\text{k}}\big)$ are the vertices of a right angled triangle.
→Show that $\text{y}=\text{A}\cos\text{x}+\text{B}\sin\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$
→In a group of $400$ people, $160$ are smokers and non-vegetarian, $100$ are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are $35\%, 20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?
→Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}'(\text{x})=\text{x}^{4}-62\text{x}^{2}+120\text{x}+9$
→$\text{f}'(\text{x})=\text{x}^{4}-62\text{x}^{2}+120\text{x}+9$