{"id":1422,"date":"2026-01-28T18:53:05","date_gmt":"2026-01-28T08:53:05","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1422"},"modified":"2026-01-28T18:53:05","modified_gmt":"2026-01-28T08:53:05","slug":"year11-math-3-2-2-introduction-to-differential-calculus-warm-up-workbook","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1422","title":{"rendered":"Year11 MATH 3-2-2 Introduction to Differential Calculus warm-up workbook"},"content":{"rendered":"\n

Unit 1<\/h4>\n\n\n\n

Focus: Limits, first principles, and basic power rule.<\/em><\/p>\n\n\n\n

    \n
  1. Evaluate lim<\/mi>x<\/mi>\u2192<\/mo>3<\/mn><\/mrow><\/msub>\u2061<\/mo>(<\/mo>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>2<\/mn>)<\/mo><\/mrow>\\lim_{x \\to 3} (x^2 – 2)<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  2. Differentiate f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>x<\/mi>5<\/mn><\/msup><\/mrow>f(x) = x^5<\/annotation><\/semantics><\/math> with respect to x<\/mi>x<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  3. Find the derivative of y<\/mi>=<\/mo>4<\/mn>x<\/mi>3<\/mn><\/msup>\u2212<\/mo>2<\/mn>x<\/mi>+<\/mo>7<\/mn><\/mrow>y = 4x^3 – 2x + 7<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  4. Differentiate f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>1<\/mn>x<\/mi>2<\/mn><\/msup><\/mfrac><\/mrow>f(x) = \\frac{1}{x^2}<\/annotation><\/semantics><\/math> (Hint: use index laws).<\/li>\n\n\n\n
  5. Use the Power Rule<\/strong> to find the gradient of y<\/mi>=<\/mo>x<\/mi>2<\/mn><\/msup><\/mrow>y = x^2<\/annotation><\/semantics><\/math> at x<\/mi>=<\/mo>3<\/mn><\/mrow>x = 3<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  6. Find d<\/mi>y<\/mi><\/mrow>d<\/mi>x<\/mi><\/mrow><\/mfrac>\\frac{dy}{dx}<\/annotation><\/semantics><\/math> if y<\/mi>=<\/mo>x<\/mi><\/msqrt><\/mrow>y = \\sqrt{x}<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  7. If f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>2<\/mn>x<\/mi>(<\/mo>x<\/mi>+<\/mo>3<\/mn>)<\/mo><\/mrow>f(x) = 2x(x + 3)<\/annotation><\/semantics><\/math>, find f<\/mi>\u2032<\/mo><\/msup>(<\/mo>x<\/mi>)<\/mo><\/mrow>f'(x)<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  8. Find the value of x<\/mi>x<\/annotation><\/semantics><\/math> where the derivative of y<\/mi>=<\/mo>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>6<\/mn>x<\/mi><\/mrow>y = x^2 – 6x<\/annotation><\/semantics><\/math> is zero.<\/li>\n\n\n\n
  9. Differentiate y<\/mi>=<\/mo>3<\/mn>x<\/mi>4<\/mn><\/msup>\u2212<\/mo>2<\/mn>x<\/mi><\/mrow>x<\/mi><\/mfrac><\/mrow>y = \\frac{3x^4 – 2x}{x}<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  10. State the formula for the derivative from first principles<\/strong>.<\/li>\n<\/ol>\n\n\n\n

    Answers:<\/strong><\/p>\n\n\n\n

      \n
    1. 7 | 2. 5<\/mn>x<\/mi>4<\/mn><\/msup><\/mrow>5x^4<\/annotation><\/semantics><\/math> | 3. 12<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>2<\/mn><\/mrow>12x^2 – 2<\/annotation><\/semantics><\/math> | 4. \u2212<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn><\/mrow><\/msup><\/mrow>-2x^{-3}<\/annotation><\/semantics><\/math> or \u2212<\/mo>2<\/mn>x<\/mi>3<\/mn><\/msup><\/mfrac><\/mrow>-\\frac{2}{x^3}<\/annotation><\/semantics><\/math> | 5. 6 | 6. 1<\/mn>2<\/mn>x<\/mi><\/msqrt><\/mrow><\/mfrac>\\frac{1}{2\\sqrt{x}}<\/annotation><\/semantics><\/math> | 7. 4<\/mn>x<\/mi>+<\/mo>6<\/mn><\/mrow>4x + 6<\/annotation><\/semantics><\/math> | 8. x<\/mi>=<\/mo>3<\/mn><\/mrow>x = 3<\/annotation><\/semantics><\/math> | 9. 9<\/mn>x<\/mi>2<\/mn><\/msup><\/mrow>9x^2<\/annotation><\/semantics><\/math> | 10. f<\/mi>\u2032<\/mo><\/msup>(<\/mo>x<\/mi>)<\/mo>=<\/mo>lim<\/mi>h<\/mi>\u2192<\/mo>0<\/mn><\/mrow><\/msub>\u2061<\/mo><\/mspace>f<\/mi>(<\/mo>x<\/mi>+<\/mo>h<\/mi>)<\/mo>\u2212<\/mo>f<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>h<\/mi><\/mfrac><\/mrow>f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) – f(x)}{h}<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n\n\n\n

      <\/p>\n","protected":false},"excerpt":{"rendered":"

      Unit 1 Focus: Limits, first principles, and basic power rule. Answers:<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1422","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1422","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1422"}],"version-history":[{"count":4,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1422\/revisions"}],"predecessor-version":[{"id":1441,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1422\/revisions\/1441"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1422"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1422"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}